






\section{Introduction and main results} \label{sec1} 
\setcounter{equation}{0}
Let $\Omega$ be a bounded and regular domain of $\mathbb{R}^N$ with $N \geq 2$. In this article we are concerned with the problem
$$ \left\{
\begin{array}{lll}
-\Delta u=\lambda(m(x)u+a(x)|u|^{p-2}u) & {\rm in } & \Omega,  \\
          \frac{\partial u}{\partial \bf{n}}=\lambda b(x)|u|^{q-2}u & {\rm on } & \partial \Omega,\\
\end{array}\right. \leqno{(P_\lambda)}
$$
where:
\begin{itemize}
\item $\Delta$ is the usual Laplacian in ${\rm I}\!{\rm R}^N$
\item $\lambda \in {\rm I}\!{\rm R}$
\item $1<q<2<p$ and if $N>2$
then $p < 2^*=\frac{2N}{N-2}$  
\item $m,a \in L^{\infty}(\Omega)$, $m^+ \not \equiv 0$
\item $b \in L^{\infty}(\partial \Omega)$
\item $\bf{n}$ is the outward unit normal to $\partial {\Omega}$. 
\end{itemize}

Our main goal in this article is to carry on the study of $(P_\lambda)$, which was addressed in \cite{RQU} for the logistic case $a \leq 0$. By variational and bifurcation techniques, we established  existence and multiplicity results for non-negative solutions of $(P_\lambda)$. Moreover, the structure of the non-negative solutions set was also discussed. We intend now to deal with these issues in the case where $a$ changes sign. 

By a solution of $(P_\lambda)$ we mean a weak solution, i.e. $u \in H^1({\Omega})$ satisfying
$$ 
\int_{\Omega} \nabla u \nabla \varphi - \lambda \int_{\Omega} m(x) u \varphi - \lambda \int_{\Omega} a(x)|u|^{p-2}u \varphi - \lambda \int_{\partial {\Omega}} b(x) |u|^{q-2}u \varphi = 0, \quad \forall \varphi \in H^1({\Omega}). 
$$
In this case, we may also say that the couple $(\lambda,u)$ is a solution of $(P_\lambda)$. As already pointed out in \cite{RQU}, solutions of $(P_\lambda)$ satisfy $u \in W^{2,r}_{\textrm{loc}}(\Omega) \cap \mathcal{C}^\theta (\overline{\Omega})$ for some $r>N$ and $0<\theta < 1$, so that by the weak maximum principle \cite[Theorem 9.1]{GT}, nontrivial non-negative solutions of $(P_\lambda)$ are strictly positive in $\Omega$.  

$(P_\lambda)$ describes the steady state of solutions of the corresponding initial boundary value problem
\begin{align}  \label{ibvp}
\begin{cases}
\dfrac{\partial u}{\partial t} = \nabla \cdot (d \nabla u) + (m(x) + a(x)|u|^{p-2})u & \mbox{in $(0, \infty)\times \Omega$}, \\ 
u(0,x) = u_0(x) \geq 0 & \mbox{in $\Omega$}, \\ 
(d\nabla u) \cdot \mathbf{n} = b(x)|u|^{q-2}u & \mbox{on $(0, \infty)\times \partial \Omega$}, 
\end{cases} 
\end{align} 
which appears as a model in population dynamics (see Cantrell and Cosner \cite{CC03}, G\'omez-Re\~nasco and L\'opez-G\'omez \cite{GRLG00}). Here the unknown function $u$ stands for the population density of some species having $m(x)$ as intrinsic growth rate and  $m(x) + a(x)|u|^{p-2}$ as extrinsic growth rate. If $a(x)\leq 0$ then the latter one is the well known logistic growth rate, with self-limitation ($a(x)<0$) or without limitation ($a(x)=0$), so that the region where $a(x)=0$ can be considered as a {\it refuge}. We can give the case $a(x)>0$ the following biological interpretation (cf. \cite{GRLG00}): in this case, the extrinsic growth rate measures the {\it symbiosis} effect due to the intraspecific cooperation whereas, in the case $a(x)<0$, it measures the crowding effect associated with competition. As for the nonlinear boundary condition, it suggests that the flux rate $(d \nabla u) \cdot \mathbf{n}$ of the population on $\partial \Omega$ is incoming or outgoing (according to the sign of $b(x)$) and depends nonlinearly on $u$ as $|u|^{q-2}u$ (cf.\ \cite{G-MM-RRS08}). From the population dynamics viewpoint, we point out that the parameter $\lambda$ appearing in $(P_\lambda)$ describes the reciprocal number of the diffusion coefficient $d>0$, and only non-negative solutions are of interest. 


Elliptic problems with indefinite nonlinearities have been studied over the last 25 years, starting with the works of Bandle, Pozio and Tesei \cite{BPT}, Ouyang \cite{Ou}, Alama and Tarantello \cite{AT}, Berestycki, Capuzzo-Dolcetta and Nirenberg \cite{BCN94, BCN95}, Lopez-Gomez \cite{LG}, etc. Brown and Zhang \cite{BZ03} and Brown \cite{B07} used the Nehari manifold (or fibering) method to discuss existence, multiplicity, and non-existence of positive solutions for the problem $$ \left\{
\begin{array}{lll}
-\Delta u=\lambda m(x)u+a(x)|u|^{p-2}u & {\rm in } & \Omega,  \\
          u=0 & {\rm on } & \partial \Omega,\\
\end{array}\right.
$$
according to the position of $\lambda$. The sublinear case $1<p<2$ and the superlinear case $p>2$ were treated in  \cite{B07} and \cite{BZ03}, respectively.  We shall see in this article that the Nehari manifold method turns out to be efficient for $(P_\lambda)$ as well.
\par 
 We note that $(P_\lambda)$ is characterized by the combination of the nonlinearities $m(x)u + a(x)|u|^{p-2}u$ in $\Omega$ and $b(x)|u|^{q-2}u$ on $\partial \Omega$. Furthermore,  the signs of $m$, $a$ and $b$ may completely change the effect of these nonlinearities. For elliptic problems with such combined nonlinearities, we refer to Chipot, Fila, and Quittner \cite{CFQ91}, L\'opez-G\'omez, M\'arquez, and Wolanski \cite{L-GMW93}, Morales-Rodrigo and Su\'arez \cite{M-RS05}, Wu \cite{W06}. Besides investigating  existence and multiplicity of positive solutions, we shall analyze as well the structure of the positive solutions set. Our approach is mainly based on a detailed study of the energy functional associated to $(P_\lambda)$. Note however that unlike most of the aforementioned works, $(P_\lambda)$ lacks coercivity on its left-hand side, since the term $(\int_\Omega |\nabla u|^2)^{\frac{1}{2}}$ does not correspond to the norm  of $u$ in $H^1(\Omega)$. 

Since the set of solutions of $(P_\lambda)$ at $\lambda = 0$ is explicitly provided by the constants, we may obtain positive solutions for $|\lambda|$ small by a bifurcation analysis on the line $(\lambda, u)=(0,c)$, where $c> 0$ is a constant, cf. \cite{U12,U13}. On the other hand, the lack of continuity of the derivative of $|u|^{q-2}u$ at $u=0$ prevents the use of the bifurcation approach to obtain positive solutions bifurcating from the zero solution. Finally, let us remark that the boundary point lemma cannot be applied directly to our problem, since it seems difficult to deduce that any nontrivial non-negative solution belongs to $\mathcal{C}^1(\overline{\Omega})$ in view of the assumption $0<q-1<1$. Therefore we are not able to infer that nontrivial non-negative solutions are positive on $\overline{\Omega}$. However we shall prove in Proposition \ref{prop:positive} that if $u$ is a nontrivial non-negative solution of $(P_\lambda)$ then the set $\{ x \in \partial \Omega : u(x) = 0 \}$ has no interior points in the relative topology of $\partial \Omega$. Note that, in the one-dimensional case $N=1$, the boundary point lemma is applicable, and we can deduce that any nontrivial non-negative weak solution of $(P_\lambda)$ is positive on $\overline{\Omega}$, see Proposition \ref{p:positivedef} in Appendix \ref{sec:posi}. 


Let us set the notations and conventions used in this article:
\begin{itemize}
\item The infimum of an empty set is assumed to be $\infty$.
\item Unless otherwise stated, for any $f \in L^1({\Omega})$ the integral $\int_\Omega f$ is considered with respect to the Lebesgue measure, whereas for any $g \in L^1(\partial {\Omega})$ the integral $\int_{\partial {\Omega}} g$  is considered with respect to the surface measure. 
\item For $r\geq 1$ the Lebesgue norm in $L^r (\Omega)$ will be denoted by $\| \cdot \|_r$ and the usual norm of $H^1(\Omega)$ by $\|\cdot \|$
\item We set $2^*=\frac{2N}{N-2}$ and $2_*=\frac{2(N-1)}{N-2}$ for $N>2$. 
\item The strong and weak convergence are denoted by $\rightarrow$ and $\rightharpoonup$, respectively. 
\item The positive
and negative parts of a function $u$ are defined by $u^{\pm} :=\max
\{\pm u,0\}$. 
\item If $U \subset {\rm I}\!{\rm R}^N$ then we denote the closure of $U$ by $\overline{U}$ and the interior of $U$ by $\text{int }U$.
\item The support of a measurable function $f$ is denoted by supp $f$.
\item If $\Phi$ is a functional defined on $H^1(\Omega)$, we set
$$\Phi^{\pm}:=\{u\in H^1(\Omega);~\Phi(u)\gtrless 0\}, \quad \Phi_{0}:= \Phi^{-1}(0) \quad \text{and} \quad \Phi_0^{\pm}:= \Phi^{\pm} \cup \Phi_{0}.$$ 
\end{itemize}


Recall that 
\begin{equation}
\lambda_1=\lambda_1(m):=\inf\left\{\int_{\Omega} |\nabla u|^2; u \in H^1(\Omega), \int_{\Omega}mu^2=1\right\}
\end{equation}
 is a principal and simple eigenvalue of the problem
$$ \left\{
\begin{array}{lll}
-\Delta u=\lambda m(x)u & {\rm in } & \Omega,  \\
          \frac{\partial u}{\partial \bf{n}}=0 & {\rm on } & \partial \Omega.\\
\end{array}\right. 
$$
It is well known (cf.\ Brown and Lin \cite{BL}) that $\lambda_1(m)>0$ if and only if $\int_{\Omega} m<0$, in which case $\lambda_1(m)$ is achieved by a unique non-constant eigenfunction $\varphi_1$ such that $\varphi_1>0$ on $\overline{\Omega}$. If $\int_{\Omega} m>0$ then $\lambda_1(m)=0$ is achieved by  $\varphi_1=\left(\int_{\Omega} m\right)^{-\frac{1}{2}}$.

We set 
\begin{equation}
\label{cpq}
C_{pq}=\frac{q(p-2)}{2(p-q)}\left(\frac{p(2-q)}{2(p-q)}\right)^{\frac{2-q}{p-2}}
\end{equation} 
and
\begin{equation}
\label{ek1}
K_1(m,a)=C_{pq} \frac{\left(\int_{\Omega} m\right)^{\frac{p-q}{p-2}}}{\left(-\int_{\Omega} a\right)^{\frac{2-q}{p-2}}},
\end{equation}
whenever $\int_\Omega m > 0>\int_\Omega a$.

Let $\varphi:[0,\infty) \rightarrow {\rm I}\!{\rm R}$ be given by \begin{equation}
\label{ephi}
\varphi (t) = t^{2-q} \int_\Omega m  + t^{p-q} 
\int_{\Omega} a + \int_{\partial \Omega} b.
\end{equation} 
It is easily seen that if either
$$\int_{\Omega} m> 0>\int_{\Omega} a \quad \text{and} \quad -K_1(m,a)<\int_{\partial \Omega} b<0$$
or
$$\int_{\Omega} m< 0<\int_{\Omega} a \quad \text{and} \quad K_1(-m,-a)>\int_{\partial \Omega} b>0$$
then $\varphi$ has two zeros $c_1<c_2$, which satisfy
\begin{equation}
\label{eph}
c_1<\left(-\frac{(2-q)\int_\Omega m}{(p-q)\int_\Omega a} \right)^{\frac{1}{p-2}} < c_2. 
\end{equation}
If either $$\int_\Omega a<0<\int_{\partial \Omega} b \quad \text{or} \quad \int_\Omega a>0>\int_{\partial \Omega} b$$
then
$\varphi$ has an unique zero, denoted by $c_0$.

In order to simplify the notation, we introduce the functionals
$$E_{\lambda}(u)=\int_{\Omega}\left(|\nabla u|^2 -\lambda m(x) u^2\right), \quad A(u)=\int_{\Omega} a(x)|u|^p \quad \text{and} \quad B(u)=\int_{\partial \Omega} b(x)|u|^q,$$
defined on $H^1({\Omega})$.
From the compactness of the Sobolev embeddings
$$H^1({\Omega})\hookrightarrow L^r({\Omega}), \quad H^1({\Omega}) \hookrightarrow L^t(\partial \Omega),$$ 
for $r \in [1,2^*)$ and $t \in [1,2_*)$ respectively, it is straightforward that $E_\lambda$ is weakly lower semi-continuous, whereas $A$ and $B$ are weakly continuous.
\begin{rem}{\rm 
 The following results, which can be easily verified, will be used repeatedly throughout this article: 
\begin{enumerate}
\item If $(u_n)$ is a bounded sequence in $H^1({\Omega})$ then we may assume that $u_n \rightharpoonup u_0$, $A(u_n) \rightarrow A(u_0)$, $B(u_n) \rightarrow B(u_0)$, and $E_\lambda(u_0) \leq \liminf E_{\lambda}(u_n)$, for some $u_0 \in H^1({\Omega})$.\\
\item If $u_n \rightharpoonup 0$ in $H^1({\Omega})$ and $\limsup E_{\lambda}(u_n) \leq 0$ then $u_n \rightarrow 0$ in $H^1({\Omega})$.
\end{enumerate}   
}\end{rem}
Let us set
\begin{equation}
\lambda_a =\lambda_a(m):=\inf\left\{\int_{\Omega} |\nabla u|^2;    u \in A_0^+, \int_{\Omega}mu^2=1 \right\},
\end{equation} 
 \begin{equation}
\lambda_b =\lambda_b(m):=\inf\left\{\int_{\Omega} |\nabla u|^2;  u \in B_0^+, \int_{\Omega}mu^2=1 \right\},
\end{equation} 
and
\begin{equation}
\label{els}
\lambda_s=\lambda_s(m,a,b)=\inf \left\{ \int_{\Omega} |\nabla u|^2;\ u \in A_0^+ \cap B_0^+, \ S(u)=1\right\},
\end{equation}
where
\begin{equation}
\label{eqs}
S(u)=\int_{\Omega} mu^2+\left(C_{pq}^{-1} B(u)A(u)^{\frac{2-q}{p-2}}\right)^{\frac{p-2}{p-q}}
\end{equation}
is defined for $u \in A_0^+ \cap B_0^+$. One may easily check that $S$ is quadratic, i.e. $S(tu)=t^2S(u)$ for $t \in {\rm I}\!{\rm R}$ and $u \in A_0^+ \cap B_0^+$. Lastly, we set
\begin{equation}
\label{defompm}
\Omega_{\pm}:=\text{int supp } a^{\pm}.
\end{equation}

We are now in position to state our main results. To begin with, we state an existence result, which provides bifurcation from zero and from infinity at $\lambda=0$ in the following sense:
\begin{defi}{\rm
It is said that {\it bifurcation from zero occurs at $\lambda = \lambda_0 \in {\rm I}\!{\rm R}$} for $(P_\lambda)$ if there exist nontrivial nonnegative solutions $u_j$ of $(P_{\lambda_j})$ such that $\lambda_j \to \lambda_0$, and $u_j \to 0$ in $\mathcal{C}(\overline{\Omega})$ as $j\to \infty$. Similarly, 
it is said that {\it bifurcation from infinity occurs at $\lambda = \lambda_\infty \in {\rm I}\!{\rm R}$} for $(P_\lambda)$ if there exist nontrivial nonnegative solutions $u_j$ of $(P_{\lambda_j})$ such that $\lambda_j \to \lambda_\infty$, and $\| u_j \|_{\mathcal{C}(\overline{\Omega})} \to \infty$ as $j\to \infty$.}
\end{defi}
\begin{thm}
\label{t1}
\strut
\begin{enumerate}
\item If $\int_{\partial \Omega} b<0$ then $\lambda_b, \lambda_s>0$. If, in addition, $b^+ \not \equiv 0$ then $(P_\lambda)$ has a nontrivial non-negative solution $u_{0,\lambda}$ for $0<\lambda <\min\{\lambda_b,\lambda_s\}$, 
which satisfies $u_{0,\lambda} \rightarrow 0$  in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^+$. 
Moreover, there exists $\lambda_n \to 0^+$ such that $\lambda_n^{-\frac{1}{2-q}}u_{0,\lambda_n} \rightarrow w_0$ in $H^1(\Omega)\cap \mathcal{C}^{\theta} (\overline{\Omega})$ as $n \to \infty$, where $w_0$ is a nontrivial non-negative solution of 
\begin{equation} \label{w0}
-\Delta w = 0 \quad\mbox{in} \ \Omega, \qquad
\frac{\partial w}{\partial \mathbf{n}} = b(x)w^{q-1} \quad \mbox{on} \
\partial \Omega.
\end{equation}
Furthermore, $w_0 > 0$ in $\Omega$, the set $\{ x\in \partial \Omega : w_0 = 0 \}$ has no interior points in the relative topology of $\partial \Omega$, and it is contained in $\{ x \in \partial \Omega : b(x) \leq 0 \}$ if $b\in \mathcal{C}(\partial \Omega)$.\\ 
	\item If $\int_{\Omega} a<0$ then $\lambda_a, \lambda_s>0$. If, in addition, $a^+ \not \equiv 0$ then $(P_\lambda)$ has a nontrivial non-negative solution $u_{2,\lambda}$ for $0<\lambda <\min\{\lambda_a,\lambda_s\}$, which satisfies $u_{2,\lambda} > 0$ in $\overline{\Omega}$ for $\lambda > 0$ sufficiently small and $\displaystyle \min_{\overline{\Omega}} u_{2,\lambda} \to \infty$ as $\lambda \to 0^+$. Moreover, there exists $\lambda_n \to 0^+$ such that $\lambda_n^{\frac{1}{p-2}}u_{2,\lambda_n} \rightarrow w_{\infty}$ in $H^1(\Omega)\cap C^\theta (\overline{\Omega})$ for some $\theta \in (0,1)$ as $n \to \infty$, where $w_\infty \in W^{2,r}(\Omega)$, $r>N$, is a solution of the  problem
	$$
	-\Delta w = a(x)w^{p-1} \quad\mbox{in} \ \Omega, \qquad
	\frac{\partial w}{\partial \mathbf{n}} = 0 \quad \mbox{on} \
	\partial \Omega. 
	$$
Furthermore $w_\infty>0$ on $\overline{\Omega}$. 
\end{enumerate}
\end{thm}


\begin{rem}
\strut
{\rm 
\begin{enumerate}
\item  Theorem \ref{t1} states that if $b^+ \not \equiv 0$ and $\int_{\partial \Omega} b<0$ then $(P_\lambda)$ has, besides the trivial branch $\{(\lambda,u)=(0,c); c \text{ is a positive constant}\}$, the solution $u_{0,\lambda}$ bifurcating from zero at $\lambda=0$. In a similar way, if  $a^+ \not \equiv 0$ and $\int_{\Omega} a<0$ then $(P_\lambda)$ has, besides the trivial branch of positive constants, the solution $u_{2,\lambda}$ bifurcating from infinity at $\lambda=0$. Furthermore, the blow up of $u_{2,\lambda}$ as $\lambda \to 0^+$ occurs uniformly on $\overline{\Omega}$. \\

\item We shall see in Proposition \ref{prop:positive} that for any $x \in \partial \Omega$ such that $b(x) > 0$ there holds $u_{0,\lambda}(x)>0$ if $\lambda>0$ is sufficiently small. More precisely, if $\Sigma \subset \partial \Omega$ is such that $\overline{\Sigma} \subset \{ x \in \partial \Omega : b(x) > 0 \}$ then $\displaystyle \inf_{\Sigma}u_{0,\lambda} > 0$ for $\lambda > 0$ sufficiently small.
\end{enumerate}
}\end{rem}



Our next results provide multiplicity of non-negative solutions and are stated under the condition
\begin{equation}
\label{cm}
\int_\Omega m >0.
\end{equation}
Since we are considering $\lambda \in {\rm I}\!{\rm R}$, the case $\int_{\Omega} m<0$ reduces to \eqref{cm} after the change of variable $\lambda \mapsto -\lambda$.

\begin{thm}
\label{t1'}
Assume \eqref{cm}, $\int_{\Omega} a<0$, and $-K_1(m,a)<\int_{\partial {\Omega}} b <0$. Then:
\begin{enumerate}
\item  $(P_\lambda)$ has a nontrivial non-negative solution $u_{1,\lambda}$ for $0<\lambda<\min\{\lambda_a,\lambda_b\}$. Moreover
$u_{1,\lambda} \rightarrow c_2$ in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^+$.\\
\item $(P_\lambda)$ has two nontrivial non-negative solutions $v_{1,\lambda}$, $v_{2,\lambda}$ for $\max\{\tilde{\lambda}_1,\tilde{\lambda}_s\}<\lambda<0$, where $\tilde{\lambda}_1=-\lambda_1(-m)$ and $\tilde{\lambda}_s=-\lambda_s(-m,-a,-b)$. Furthermore, $v_{1,\lambda} \to c_1$ and $v_{2,\lambda} \to c_2$ in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^-$.
\end{enumerate}

\end{thm}



      






Under the assumptions of Theorem \ref{t1'} and stronger regularity conditions on $m,a,b$, we shall obtain, for $\lambda > 0$ sufficiently small, a positive solution of $(P_\lambda)$ converging to $c_1$ as $\lambda\to 0^+$. This will be carried out by a bifurcation argument and, as a consequence, will provide at least four nontrivial non-negative solutions of $(P_\lambda)$ for $\lambda > 0$ small enough. Since this argument requires only the existence of zeros of $\varphi$ and some regularity on $m$, $a,$ and $b$, we shall assume, in addition to \eqref{cm},
\begin{align} \label{assump:mbreg}
m,a \in \mathcal{C}^\alpha (\overline{\Omega}) \quad \text{and} \quad b \in 
\mathcal{C}^{1+\alpha}(\partial \Omega),\quad \text{for some} 
\quad 0<\alpha < 1,
\end{align}
and
\begin{align}
\int_\Omega a<0 \quad \text{and} \quad -\tilde{K}_1(m,a)< \int_{\partial \Omega} b<0, \label{assump:tildK}
\end{align}
where 
\begin{equation}
\label{ktilda}
\tilde{K}_1(m,a) = \tilde{C}_{pq} \frac{\left(\int_{\Omega} m\right)^{\frac{p-q}{p-2}}}{\left(-\int_{\Omega} a\right)^{\frac{2-q}{p-2}}},
\end{equation}
and $$\tilde{C}_{pq} =\left(\frac{q}{2}\left( \frac{p}{2} \right)^{\frac{2-q}{p-2}} \right)^{-1}C_{pq}.$$ It is easily seen that $\tilde{C}_{pq} >C_{pq}$, so that  $\tilde{K}_1(m,a)>K_1(m,a)$.  

Hereafter, by a classical positive solution of $(P_\lambda)$ we mean $u \in \mathcal{C}^{2+\theta}(\overline{\Omega})$ for some $0< \theta < 1$ which satisfies $(P_\lambda)$ in the classical sense and is strictly positive on $\overline{\Omega}$.

\begin{thm} \label{t3b} 
Assume \eqref{assump:mbreg}. If $a$ changes sign and $\int_\Omega a < 0$, then, for $\lambda>0$ sufficiently small, the solution $u_{2,\lambda}$ obtained in Theorem \ref{t1} is a classical positive solution of $(P_\lambda)$, which is moreover unstable.
\end{thm}

\begin{rem}
\label{rw0}
{\rm Since $1<q<2$, solutions of \eqref{w0} are in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ with $\theta \in (0,1)$, so that the boundary point lemma is not applicable, and consequently we can not deduce the positivity of $w_0$ on $\partial \Omega$. If this is the case then we can prove a result similar to Theorem \ref{t3b} for $u_{0,\lambda}$, obtained in Theorem \ref{t1}. More precisely, let us assume \eqref{assump:mbreg} and $w_0>0$ on $\overline{\Omega}$. If $b$ changes sign and $\int_{\partial \Omega}b<0$ then, for $\lambda > 0$ sufficiently small, $u_{0,\lambda}$ is a classical positive solution of $(P_\lambda)$ which is moreover asymptotically stable. We include a sketch of the proof of this result in Section \ref{sec5}.
}\end{rem}




\begin{thm} \label{t3} 
Assume \eqref{cm}, \eqref{assump:mbreg} and \eqref{assump:tildK}.
Then there exists $\underline{\lambda} >0$ such that: 
\begin{enumerate}
  \item  $u_{1,\lambda}$ is a classical positive solution of $(P_\lambda)$ which is asymptotically stable for $\lambda \in (0,\underline{\lambda})$. Moreover, $(P_\lambda)$ has a classical positive solution $u_{3,\lambda}$ for $\lambda \in (0,\underline{\lambda})$ which is unstable. These solutions are continuous in $\mathcal{C}^{2+\alpha}(\overline{\Omega})$ with respect to $\lambda$ and emanate from $(0,c_2)$ and $(0,c_1)$ respectively, i.e. $u_{1,0}=c_2$ and $u_{3,0}=c_1$. Finally, there are no other classical positive solutions of $(P_\lambda)$ converging to some positive constant in $\mathcal{C}(\overline{\Omega})$ as $\lambda \to 0^+$. \\
  \item $v_{1,\lambda}, v_{2,\lambda}$ are classical positive solutions of $(P_\lambda)$ for $\lambda \in (-\underline{\lambda},0)$, which are asymptotically stable, and unstable, respectively. Moreover, these solutions are continuous in $\mathcal{C}^{2+\alpha}(\overline{\Omega})$ with respect to $\lambda$ and emanate from $(0,c_1)$ and $(0,c_2)$ respectively, i.e. $v_{1,0}=c_1$ and $v_{2,0}=c_2$. Finally, there are no other classical positive solutions of $(P_\lambda)$ converging to some positive constant in  $\mathcal{C}(\overline{\Omega})$ as $\lambda \to 0^-$. 
\end{enumerate}
\end{thm}


\begin{rem}
{\rm Theorem \ref{t3} states, in particular, that $v_{1,\lambda}$, $v_{2,\lambda}$ are the extensions of $u_{3,\lambda}$, $u_{1,\lambda}$, respectively, to the region $\lambda < 0$ (see Figure \ref{fig1}).}
\end{rem} 




Combining Theorems \ref{t1} and \ref{t1'}, we get a multiplicity result for $(P_\lambda)$ with $\lambda > 0$ small, which is obtained by  variational techniques:


\begin{cor}  \label{t2}
Under the assumptions of Theorem \ref{t1'}, assume moreover that $a^+\not\equiv 0$ and $b^+\not\equiv 0$.
Then $(P_\lambda)$ has three nontrivial non-negative solutions $u_{j,\lambda}$, $j=0,1,2$, for $0<\lambda <\min\{\lambda_s,\lambda_a,\lambda_b\}$, which satisfy $u_{0,\lambda} \rightarrow 0$ and $u_{1,\lambda} \rightarrow c_2$ in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ and $\displaystyle \min_{\overline{\Omega}} u_{2,\lambda} \to \infty$ as $\lambda \to 0^+$.
\end{cor}       

From Corollary \ref{t2} and Theorem \ref{t3}(1), we infer the following multiplicity result, which is obtained combining the variational and bifurcation approaches: 

\begin{cor}
In addition to the assumptions of Theorem \ref{t1'}, we assume \eqref{assump:mbreg}. 
Assume moreover $a^+\not \equiv 0$. Then $(P_\lambda)$ has at least  three classical positive solutions for $0<\lambda<\underline{\lambda}$, namely $u_{j,\lambda}$, $j=1,2,3$.  If, in addition, $b^+ \not \equiv 0$ then $(P_\lambda)$ has a fourth nontrivial non-negative solution for $0<\lambda<\underline{\lambda}$, namely, $u_{0,\lambda}$.
\end{cor}



\begin{rem} \label{rem:classifiP}
{\rm We shall see in Proposition \ref{prop:bif} that under \eqref{cm},  \eqref{assump:mbreg}, and \eqref{assump:tildK}, $(P_\lambda)$ has, for $|\lambda|$ sufficiently small, two classical positive solutions $U_{j, \lambda}$, $j=1,2$, which are continuous (with respect to $\lambda$) in $\mathcal{C}^{2+\alpha}(\overline{\Omega})$ and satisfy $U_{j,0}=c_j$, $j=1,2$. Furthermore, this result does not require the condition $p<2^*$.
So, under \eqref{cm}, \eqref{assump:mbreg} and the conditions
$$a,b \text{ change sign,} \quad \int_\Omega a<0, \quad \text{and} \quad  \int_{\partial \Omega} b<0,$$ 
 Theorem \ref{t1} and Proposition \ref{prop:bif} provide, according to the value of $\int_{\partial \Omega}b$,  the following result on the number of nontrivial non-negative solutions of $(P_\lambda)$:\\
\begin{enumerate}
\item If $\int_{\partial \Omega}b >-\tilde{K_1}(m,a)$ then $(P_\lambda)$ has at least four nontrivial non-negative solutions (two variational nontrivial non-negative solutions, among which one is a classical positive solution, and two bifurcating classical positive solutions) for $\lambda > 0$ sufficiently small.\\
\item If $\int_{\partial \Omega}b \leq -\tilde{K_1}(m,a)$ then $(P_\lambda)$ has at least two (variational) nontrivial non-negative solutions, among which one is a classical positive solution, for $\lambda > 0$ sufficiently small. Moreover, if $\int_{\partial \Omega}b < -\tilde{K_1}(m,a)$ then $(P_\lambda)$ has no classical positive solutions converging to a positive constant as $\lambda \to 0^+$.
\end{enumerate} 
}\end{rem}


The following condition provides an {\it a priori} bound on the values of $\lambda$ for which $(P_\lambda)$ has a nontrivial non-negative solution: 
	$$ \left\{
	\begin{array}{lll} \text{There are smooth sub-domains }D_{\pm} \Subset \Omega \text{ such that }\\ D_{\pm} \subset \Omega_{\pm}
	 \text{ and $m$ changes sign in $D_+$ and in $D_-$.} \end{array}\right. \leqno{\mathcal{(H)}}$$


	\begin{thm} 
	\label{t4}
	Assume that $a$ changes sign and $\mathcal{(H)}$ holds.  Then there exists a constant $\Lambda > 0$ such that if $u$ is a nontrivial non-negative solution of $(P_\lambda)$ then $|\lambda|\leq \Lambda$. 
	\end{thm}



As a particular case of $(P_\lambda)$, we shall consider the problem
$$ \left\{
\begin{array}{lll}
-\Delta u=\lambda m(x)(u-|u|^{p-2}u) & {\rm in } & \Omega,  \\ 
          \frac{\partial u}{\partial \bf{n}}=\lambda b(x)|u|^{q-2}u & {\rm on } & \partial \Omega,\\
\end{array}\right. \leqno{(Q_\lambda)} 
$$
where $\lambda$, $m$, $b$, $p$, $q$ are as above. Note that $(Q_\lambda)$ corresponds to $(P_\lambda)$ with $a\equiv -m$.  The nonlinearity $\lambda m(x)(u-|u|^{p-2}u)$ arises from population genetics and has already been studied under homogeneous boundary conditions in \cite{BH, F, S}.


Let 
\begin{equation*}
\lambda_r =\lambda_r(m,b):=\inf\left\{\int_{\Omega} |\nabla u|^2; \ u \in H^1(\Omega), \ \int_{\Omega}mu^2=1, \int_{\Omega} m|u|^p \leq 0 \leq \int_{\partial {\Omega}} b|u|^q  \right\},
\end{equation*} 
$$\lambda_t=\lambda_t(m,b):=\inf \left\{ \int_{\Omega} |\nabla u|^2;\ u \in H^1(\Omega), \ \int_{\Omega} m|u|^p \leq 0 \leq \int_{\partial {\Omega}} b|u|^q, \ T(u)=1\right\},$$
where
$$T(u)=\int_{\Omega} mu^2+\left(C_{pq}^{-1} \left(\int_{\partial {\Omega}} b|u|^q \right)\left(-\int_{\Omega} m|u|^p\right)^{\frac{2-q}{p-2}}\right)^{\frac{p-2}{p-q}},$$
is defined for $u \in H^1({\Omega})$ such that $\int_{\Omega} m|u|^p \leq 0 \leq \int_{\partial {\Omega}} b|u|^q$. Note that $\lambda_r=\min\{\lambda_a(m),\lambda_b(m)\}$ and $\lambda_t=\lambda_s(m,a,b)$ with $a\equiv -m$. 

Finally, let 
$$\phi(t)=\left(t^{2-q}-t^{p-q}\right)\int_{\Omega} m + \int_{\partial {\Omega}} b.$$

The assumption of Theorem \ref{t1'}
reads now
\begin{equation}
\label{emf}\int_{\Omega} m>0 >\int_{\partial \Omega} b> -C_{pq} \int_{\Omega} m,
\end{equation}
in which case $\phi$ has two positive zeros $c_1<c_2$. 





Applying Theorems \ref{t1} and \ref{t1'} to $(Q_\lambda)$ we obtain the following: 

\begin{thm}
\label{t5}
Assume \eqref{emf} and $m$ changes sign. Then $\lambda_r,\lambda_t>0$ and:
\begin{enumerate}
\item $(Q_\lambda)$ has two nontrivial non-negative solutions $u_{1,\lambda}$, $u_{2,\lambda}$ for $0<\lambda< \min\{\lambda_1, \lambda_t\}$, which satisfy $u_{1,\lambda} \rightarrow c_2$ in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ and $\displaystyle \min_{\overline{\Omega}} u_{2,\lambda} \rightarrow \infty$ as $\lambda \to 0^+$. If, in addition, $b^+ \not \equiv 0$, then $(Q_\lambda)$ has a further nontrivial non-negative solution $u_{0,\lambda}$ for $0<\lambda <\lambda_r$,
which satisfies $u_{0,\lambda} \rightarrow 0$  in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ as $\lambda \to 0^+$. \\
\item $(Q_\lambda)$ has two nontrivial non-negative solutions $v_{1,\lambda}$, $v_{2,\lambda}$ for $\max\{\tilde{\lambda}_1,\tilde{\lambda}_t\}<\lambda<0$, where $\tilde{\lambda}_1=-\lambda_1(-m)$ and $\tilde{\lambda}_t=-\lambda_t(-m,-b)$. Furthermore, $v_{1,\lambda} \to c_1$ and $v_{2,\lambda} \to c_2$ in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^-$.
\end{enumerate}
\end{thm}      

Lastly, we consider the case where $\phi$ has a unique positive zero $c_0$ such that $\phi'(c_0)=0$, which occurs precisely when
\begin{align} \label{assump:=-tildeK1}
\int_\Omega m > 0 > \int_{\partial \Omega}b = - \tilde{K}_1(m,-m),
\end{align}
where $\tilde{K}_1$ is defined in \eqref{ktilda}.
Under (\ref{assump:mbreg}), we obtain a smooth curve of classical positive solutions of $(Q_\lambda)$ for $|\lambda|$ sufficiently small by a bifurcation argument. 







The following result asserts that if \eqref{assump:=-tildeK1} holds then $(0, c_0)$ is a turning point to the right on the smooth curve of positive solutions of $(Q_\lambda)$. 

\begin{thm} \label{thm:a=-m:bif} 
Assume \eqref{assump:mbreg}. If $(Q_\lambda)$ has a classical positive solution $u$ with $\lambda \not= 0$ such that $u_\lambda \rightarrow c$ in $\mathcal{C}(\overline{\Omega})$ as $\lambda \to 0$, where $c$ is a positive constant, then  $\phi (c) = 0$. Moreover: 
\begin{enumerate}
\item Assume \eqref{assump:tildK} with $a=-m$. Then  there exist two arbitrarily smooth maps $\lambda \mapsto U_{1,\lambda}, U_{2,\lambda} \in \mathcal{C}^{2+\alpha}(\overline{\Omega})$ for $\lambda$ close to $0$ such that $U_{1,\lambda}$, $U_{2,\lambda}$ are classical positive solutions of $(Q_\lambda)$ and satisfy $U_{1,0}=c_1$ and $U_{2,0}=c_2$. \\


\item Assume \eqref{assump:=-tildeK1}.
Then there exist two arbitrarily smooth maps $t \mapsto \lambda(t) \in {\rm I}\!{\rm R}$ and $t \mapsto u(t)\in \mathcal{C}^{2+\alpha}(\overline{\Omega})$ for $t$ close to $c_0$ such that $(\lambda,u)=(\lambda(t),u(t))$ is a classical positive solution of $(Q_\lambda)$ with $\lambda(c_0)=\lambda'(c_0)=0$, $\lambda''(c_0)>0$, and $u(t)= t + o(1)$ as $t\to c_0$. Moreover, there exists a constant $\varepsilon > 0$ such that $u(t)$ is asymptotically stable for $c_0<t<c_0+\varepsilon$ and unstable for $c_0-\varepsilon < t < c_0$. 
\end{enumerate}
\end{thm} 

\begin{rem}{\rm \strut
\begin{enumerate}
\item Theorem \ref{thm:a=-m:bif} holds in fact without the condition $p<2^*$. 
\item Assertion (1) in Theorem \ref{thm:a=-m:bif} corresponds to Theorem \ref{t3} for $(Q_\lambda)$. 
\end{enumerate}

}\end{rem}





\begin{rem}\label{rem:classifiQ}
{\rm 
 The results on the number of nontrivial non-negative solutions from Remark \ref{rem:classifiP} can be sharpened for $(Q_\lambda)$. In this case, Theorems \ref{t1} and \ref{thm:a=-m:bif} provide that, under \eqref{assump:mbreg}, if $m$ and $b$ change sign, and $\int_\Omega m>0>    \int_{\partial \Omega} b$, then $(Q_\lambda)$ has: 
  \begin{enumerate}
  \item at least four nontrivial non-negative solutions (two variational nontrivial non-negative solutions, among which one is a classical positive solution, and two bifurcating classical positive solutions) for $\lambda>0$ sufficiently small if $\int_{\partial \Omega}b \geq -\tilde{K_1}(m,-m)$.  \\
  \item at least two (variational) nontrivial non-negative solutions, among which one is a classical positive solution, for $\lambda>0$ sufficiently small and no classical positive solutions converging to a positive constant as $\lambda \to 0^+$ if $\int_{\partial \Omega}b < -\tilde{K_1}(m,-m)$. 
  \end{enumerate}
}\end{rem}      


\subsection{Suggested bifurcation diagrams}
\strut
\medskip

In view of the results stated above, we analyze now the possible bifurcations diagrams of $(P_\lambda)$ and $(Q_\lambda)$. This will be done assuming that $a$ changes sign and $(\mathcal{H})$ holds, in which case Theorem \ref{t4} ensures that $(P_\lambda)$ has no nontrivial non-negative solution for  $|\lambda|>\Lambda$.


\begin{enumerate}
\item Assume \eqref{cm}, $a$ and $b$ change sign, and $\int_\Omega a<0$. If $-K_1(m,a)<\int_{\partial {\Omega}} b<0$ then, by Theorems \ref{t1}, \ref{t1'}, and \ref{t3}, the bifurcation diagram for $(P_\lambda)$ is suggested by Figure \ref{fig1}.
Moreover, Remark \ref{rem:classifiP} suggests that this bifurcation diagram  approaches the one shown in Figure \ref{figbinfty} as $\int_{\partial \Omega} b \to -\infty$. Indeed, note that the value $\Lambda$ provided by Theorem \ref{t4} does not depend on $b$. Furthermore, it can be shown that $\lambda_b$ and $\lambda_s$ stay bounded away from zero if $b^+$ is bounded from above (cf. Remark \ref{rbb}). By a formal observation, the nonlinear boundary condition in $(P_\lambda)$ approaches the Dirichlet boundary condition $u=0$
as $\int_{\partial \Omega}b \to -\infty$. So, after the change of variable $v = \lambda^{\frac{1}{p-2}}u$ for $u\geq 0$, the limiting problem for $(P_\lambda)$ when $\int_{\partial \Omega}b \to -\infty$ would be $$-\Delta v = \lambda m(x)v + a(x)v^{p-1} \mbox{ in } \Omega, \quad v=0  \mbox{ on } \partial \Omega.$$ This problem has been investigated by Ouyang \cite{Ou}  in the case $m \equiv 1$ and the existence of a single turning point in the positive solutions set has been proved under some conditions on $a$. We expect then that the bifurcation diagram of $(P_\lambda)$ has a unique turning point if $\int_{\partial \Omega}b \ll 0$.
\\

\item Assume \eqref{cm}, $b \leq 0$, $a$ changes sign, and $\int_{\Omega} a<0$.\\
\begin{enumerate}
\item If $-K_1(m,a)<\int_{\partial {\Omega}} b<0$ then the bifurcation diagram for $(P_\lambda)$ in the case $b\leq 0$ is suggested by Figure \ref{fig0sol}. This is motivated by Theorems \ref{t1}, \ref{t1'}, and \ref{t3}, and the fact that $u_{0,\lambda}$ approaches zero as $b^+$ converges to zero. More precisely, in Proposition \ref{pmn1} we prove that if $b_n \rightarrow b$ in $L^{\infty}(\partial \Omega)$ with $b_n^+ \not \equiv 0$ for every $n$, then there exists $\lambda_0>0$ such that the solution $u_{0,\lambda,b_n}$ exists for every $n$, and satisfies $u_{0,\lambda,b_n} \to 0$ in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ and $0<\lambda<\lambda_0$. Note that in \cite{RQU3} we have obtained bifurcation from zero in the case $b\leq 0$ and $b\not\equiv 0$ when $a< 0$.\\
\item In a similar way, if $-\tilde{K}_1(m,a)-\varepsilon<\int_{\partial {\Omega}} b<-\tilde{K}_1(m,a)$ with $\varepsilon>0$ sufficiently small then the bifurcation diagram for $(P_\lambda)$ would be suggested by Figure \ref{figbnonposi:large}. Lastly, as $\int_{\partial {\Omega}} b \to -\infty$, the bifurcation diagram for $(P_\lambda)$ is expected to approach Figure \ref{figbnonposi:enoughlarge}.\\
\end{enumerate}
\item Assume that $m$ and $b$ change sign, and $\int_\Omega m>0>    \int_{\partial \Omega} b = -\tilde{K_1}(m,-m)$. By Remark \ref{rem:classifiQ}, the bifurcation diagram of $(Q_\lambda)$ is suggested by Figure \ref{fig:a=-m}.
\end{enumerate}

\begin{rem} \label{rem:bio} {\rm
From the biological viewpoint, the bifurcation diagram 
suggested in Figure \ref{fig1} would provide three possible conditional states for the population as the diffusion rate grows to infinity, namely: extinction, explosion and persistence with a spatially uniform distribution.
}\end{rem} 

	\begin{figure}[H]
		\input fig14_0516ss
	  \caption{Suggested bifurcation diagram for $(P_\lambda)$ when $a$ and $b$ change sign, $\int_\Omega m > 0 > \int_\Omega a$, and $0> \int_{\partial \Omega}b > -K_1(m,a)$.} \label{fig1}
	    \end{figure} 
	   
	\begin{figure}[H]
	\input fig14_0727s
	  \caption{Suggested bifurcation diagram for $(P_\lambda)$ when $a$ and $b$ change sign, 
	$\int_\Omega m > 0 > \int_\Omega a$,  and $\int_{\partial \Omega}b \to -\infty$.}
	\label{figbinfty}
	 \end{figure}

	\begin{figure}[H]
	\input fig14_0702s
  \caption{Suggested bifurcation diagram for $(P_\lambda)$ when $a$ changes sign, $b \leq 0$, $\int_\Omega m > 0 > \int_\Omega a$, and  $0> \int_{\partial \Omega}b > -K_1(m,a)$.} \label{fig0sol}
    \end{figure} 

\begin{figure}[H]
\input fig14_0816s
  \caption{Suggested bifurcation diagram for $(P_\lambda)$ when $a$ changes sign, $b\leq 0$, $\int_\Omega m > 0 > \int_\Omega a$, and $\int_{\partial
\Omega}b < -\tilde{K}_1(m,a)$.}
\label{figbnonposi:large} 
 \end{figure}

\begin{figure}[H]
      \begin{center}
   
      \subfigure[Bifurcation diagram with a single turning point.]{
    %
    %
        \input fig14_0816b02s
    %
          \label{fig:turningpt}
      }
      %
      \hfill
     
      \subfigure[Bifurcation diagram without a turning point.]{
    %
    %
          \input fig14_0816c02s
        \label{fig:noturningpt}
      }
      \end{center}
    %
      \caption{Suggested bifurcation diagram for $(P_\lambda)$ when $a$ changes sign, $b\leq 0$, $\int_\Omega m > 0 > \int_\Omega a$, and $\int_{\partial
\Omega}b \to -\infty$.}
      \label{figbnonposi:enoughlarge}
    \end{figure} 
	
\begin{figure}[H]
	\input fig14_0723s
  \caption{Suggested bifurcation diagram for $(Q_\lambda)$ when $m$ and $b$ change sign and $\int_\Omega m > 0 > \int_{\partial \Omega}b = -\tilde{K}_1(m,-m)$.} \label{fig:a=-m} 
    \end{figure} 



The rest of this article is organized as follows: in Section \ref{sec2} we collect some preliminary results on $\lambda_a$, $\lambda_b$, $\lambda_s$ and $\mathcal{N}_\lambda$, the Nehari manifold associated to $(P_\lambda)$. In Section \ref{sec3}, we obtain two minimizers for $I_\lambda$ (the energy functional associated to $(P_\lambda)$) constrained to $\mathcal{N}_\lambda$. These minimizers are located in $\mathcal{N}_\lambda^+$ and correspond to local minimizers of $I_\lambda$. In Section \ref{sec4}, we obtain a further minimizer of $I_\lambda$ constrained to $\mathcal{N}_\lambda^-$, which corresponds to a minimax critical point of $I_\lambda$. Finally, in Section \ref{sec5}, we carry out a bifurcation analysis of $(P_\lambda)$ and prove the theorems stated above.



\section{Preliminaries} \label{sec2} 
Solutions of $(P_\lambda)$ shall be obtained as critical points of the functional
$$I_{\lambda}(u)=\frac{1}{2}E_{\lambda}(u) -\frac{\lambda}{p}A(u) -\frac{\lambda}{q}B(u),$$
defined on $H^1(\Omega)$. 
It is straightforward that $I_{\lambda}$ is weakly lower semicontinuous for any $\lambda \in {\rm I}\!{\rm R}$. In contrast with the case $a < 0$, for any value of $\lambda$ the functional $I_\lambda$ is not coercive. Consequently we shall restrict $I_\lambda$ to its associated Nehari manifold, which is given by
\begin{eqnarray*}
\mathcal{N}_\lambda &=&\{u \in H^1(\Omega)\setminus \{0\};\, \langle I'_{\lambda}(u), u \rangle=0\}\\
&=&\left\{u \in H^1(\Omega)\setminus \{0\}; \, E_\lambda(u)=\lambda \left[A(u)+B(u)\right]\right\}.
\end{eqnarray*}
$\mathcal{N}_\lambda$ is naturally split into $\mathcal{N}_\lambda^+$, $\mathcal{N}_\lambda^-$ and $\mathcal{N}_\lambda^0$, given by
$$\mathcal{N}_{\lambda}^{\pm}=\left\{u\in \mathcal{N}_\lambda; \,
\langle J'_\lambda(u), u \rangle \gtrless 0\right\} \quad \text{and} \quad 
\mathcal{N}_\lambda^0=\left\{u\in \mathcal{N}_\lambda; \,
\langle J'_\lambda(u), u \rangle = 0\right\},$$
where $J_\lambda(u)=\langle I'_\lambda(u),u \rangle$ for $u \in H^1(\Omega)$.
Thus
$$
\mathcal{N}_{\lambda}^{\pm}
=\left\{u\in \mathcal{N}_\lambda; \,
E_\lambda(u)\lessgtr \lambda \frac{p-q}{p-2}B(u)\right\}
=\left\{u\in \mathcal{N}_\lambda; \,
E_\lambda(u)\gtrless \lambda \frac{p-q}{2-q}A(u)\right\}.
$$
and
$$\mathcal{N}_\lambda^0
=\left\{u\in \mathcal{N}_\lambda; \,
E_\lambda(u)= \lambda \frac{p-q}{p-2}B(u)=\lambda \frac{p-q}{2-q}A(u)\right\}.$$


It is well-known that $\mathcal{N}_\lambda \setminus \mathcal{N}_\lambda^0$ is a $\mathcal{C}^1$ manifold defined by a natural constraint, i.e. 
any critical point of the restriction of $I_{\lambda}$ to $\mathcal{N}_\lambda \setminus \mathcal{N}_\lambda^0$ is a critical point of $I_{\lambda}$ (see for instance \cite[Theorem 2.3]{BZ03}).
Given $u\in H^1(\Omega) \setminus \{0\}$, we set $$j_u(t)=I_{\lambda}(tu), \quad t>0.$$
Then $tu \in \mathcal{N}_\lambda^{\pm}$ if, and only if,  $j'_u(t)=0$ and  $j''_u(t) \gtrless 0$. We shall use the map $j_u$ to deduce some properties of $\mathcal{N}_\lambda$. 
Note that $j_u(t)=t^qi_u(t)$,
where 
\begin{equation}
\label{eqi}
i_u(t)=\frac{1}{2}t^{2-q} E_\lambda(u)-\frac{\lambda}{p}t^{p-q}A(u)-\frac{\lambda}{q}B(u)
\end{equation}
for $t>0$. 



\begin{rem}
\label{cn}{\rm 
It is easily seen that $c$ is a positive zero of $\varphi$ if and only if $c \in \mathcal{N}_\lambda$, for any $\lambda \in {\rm I}\!{\rm R}$. Additionally, if $\varphi'(c) \gtrless 0$ then $c \in \mathcal{N}_\lambda^{\pm}$. More precisely:
\begin{enumerate}
\item  If $\int_{\Omega} m> 0>\int_{\Omega} a$ and $-K_1(m,a)<\int_{\partial \Omega} b<0$ then $c_1 \in \mathcal{N}_\lambda^-$ and $c_2 \in \mathcal{N}_\lambda^+$.
\item If $\int_{\Omega} m< 0<\int_{\Omega} a$ and $0<\int_{\partial \Omega} b<K_1(-m,-a)$ then $c_1 \in \mathcal{N}_\lambda^+$ and $c_2 \in \mathcal{N}_\lambda^-$.
\item If $\int_{\Omega} a> 0> \int_{\partial \Omega} b$  then $c_0 \in \mathcal{N}_\lambda^-$.
\item If $\int_{\Omega} a< 0< \int_{\partial \Omega} b$ then $c_0 \in \mathcal{N}_\lambda^+$.
\end{enumerate}
}\end{rem}

The role of $\lambda_s$ in the study of $\mathcal{N}_\lambda$ becomes clear in the next result:

\begin{prop}
\label{p1}
If either $\int_{\Omega} a<0$ or $\int_{\partial \Omega} b<0$ or
\begin{equation}
\label{n1}
\int_{\Omega} m<0<\int_{\Omega} a \quad \text{and} \quad 0<\int_{\partial \Omega} b<K_1(-m,-a).
\end{equation}
then $\lambda_s$, given by \eqref{els}, is achieved. In particular, $\lambda_s>0$.
Furthermore, if $\lambda \in (0,\lambda_s)$ then for every $u \in B^+ \cap E_{\lambda}^+ \cap A^+$ there are $0<t_1<t_2$ such that $t_1 u \in \mathcal{N}_\lambda^+$ and $t_2 u \in \mathcal{N}_\lambda^-$, and there is no other $t>0$ such that $tu \in \mathcal{N}_\lambda$.
\end{prop}


\begin{proof}
Let $(u_n)$ be a minimizing sequence for $\lambda_s$, i.e.
$$A(u_n) \geq 0, \quad  B(u_n)\geq 0, \quad S(u_n)=1 \quad \text{and} \quad \int_{\Omega} |\nabla u_n|^2 \rightarrow \lambda_s.$$
If $(u_n)$ is unbounded then we set $v_n=\frac{u_n}{\|u_n\|}$. We may assume that 
$$v_n \rightharpoonup v_0, \quad A(v_n) \rightarrow A(v_0), \quad \text{and} \quad B(v_n) \rightarrow B(v_0).$$
Thus 
$$A(v_0) \geq 0, \quad B(v_0) \geq 0, \quad S(v_0)=0, \quad \text{and} \quad \int_{\Omega} |\nabla v_n|^2 \rightarrow 0.$$ Hence $v_0$ is a nonzero constant, so that 
$$\int_{\Omega} a\geq 0, \quad \int_{\partial \Omega} b\geq 0 \quad \text{and} \quad \int_{\Omega} m +\left(C_{pq}^{-1} \left(\int_{\partial \Omega} b\right) \left(\int_{\Omega} a\right)^{\frac{2-q}{p-2}}\right)^{\frac{p-2}{p-q}}=0,$$
which contradicts our assumption.
Thus $(u_n)$ is bounded and we may assume that $u_n \rightharpoonup u_0$. If $\lambda_s=0$ then $u_0$ is a nonzero constant, and from $A(u_0)\geq 0$, $B(u_0)\geq 0$ and $S(u_0)=1$ we get
$$\int_{\Omega} a\geq 0, \quad \int_{\partial \Omega} b\geq 0 \quad \text{and} \quad \int_{\Omega} m +\left(C_{pq}^{-1} \left(\int_{\partial \Omega} b\right) \left(\int_{\Omega} a\right)^{\frac{2-q}{p-2}}\right)^{\frac{p-2}{p-q}}>0,$$
which contradicts again our assumption.
Therefore $\lambda_s=\int_{\Omega} |\nabla u_0|^2>0$.

Now, let $\lambda \in (0,\lambda_s)$ and $u \in A^+ \cap B^+ \cap E_{\lambda}^+$. Then $j_u$ has two critical points if $j_u(t)>0$ for some $t>0$. In this case,  $j_u$ has a local minimum and a global maximum, i.e. there are $t_1<t_2$ such that $t_1 u \in \mathcal{N}_{\lambda}^+$ and $t_2 u \in \mathcal{N}_{\lambda}^-$. Note that $j_u(t)>0$ if and only if $i_u(t)>0$, where $i_u$ is given by \eqref{eqi}. One may easily check that $i_u$ has a global maximum point given by 
\begin{equation}
\label{eqto}
t_0(u)=\left(\frac{p(2-q)}{2(p-q)} \frac{E_\lambda(u)}{\lambda A(u)}\right)^{\frac{1}{p-2}}.
\end{equation}
We have
\begin{eqnarray*}
i_u(t_0(u))>0 &\Leftrightarrow&
\lambda B(u)<C_{pq} \frac{E_\lambda(u)^{\frac{p-q}{p-2}}}{\left(\lambda A(u)\right)^{\frac{2-q}{p-2}}} \\ 
&\Leftrightarrow &
B(u)<C_{pq} \frac{\left(\lambda^{-1}\int_{\Omega} |\nabla u|^2 -\int_{\Omega} mu^2\right)^{\frac{p-q}{p-2}}}{A(u)^{\frac{2-q}{p-2}}}\\
&\Leftrightarrow &
E_{\lambda}(u)-\lambda \left(C_{pq}^{-1} B(u)A(u)^{\frac{2-q}{p-2}}\right)^{\frac{p-2}{p-q}}>0\\
&\Leftrightarrow & \int_{\Omega} |\nabla u|^2 -\lambda S(u)>0,
\end{eqnarray*}
where $S(u)$ is defined in \eqref{eqs}. Since $$\lambda <\lambda_s=\inf \left\{ \int_{\Omega} |\nabla u|^2;\ u \in A_0^+ \cap B_0^+, \ S(u)=1\right\},$$ we get the existence of $t_1<t_2$.
Finally, from the expression of $j_u$ it is easily seen that $j_u$ can not have more than two critical points.
\end{proof}

We shall get now some kind of local coercivity for $E_\lambda$. More precisely, we shall prove that $E_\lambda$ is coercive on $A_0^+$  (respect. $B_0^+$) for $\lambda<\lambda_a$ (respect. $\lambda<\lambda_b$). To this end, we deal with the maps $\alpha_1,\beta_1:{\rm I}\!{\rm R} \mapsto {\rm I}\!{\rm R}$ given by

\begin{equation}
\alpha_1(\lambda)=\inf\{E_\lambda(u); \ u \in A_0^+,\ \|u\|_2=1\},
\end{equation}

\begin{equation}
\beta_1(\lambda)=\inf\{E_\lambda(u); \ u \in B_0^+,\ \|u\|_2=1\}.
\end{equation}
 
Note that if $\alpha_1(\lambda)>0$ (respect. $\beta_1(\lambda)>0$) then $A_0^+ \setminus \{0\} \subset E_\lambda^+$ (respect $B_0^+ \setminus \{0\} \subset E_\lambda^+$).
From \cite{RQU}, we have the following results:

\begin{prop}
\label{pal1}
\strut
\begin{enumerate}
\item $\alpha_1$ and $\beta_1$ are concave (and therefore continuous).
\item $\lambda_b>0$ if and only if either $\int_{\Omega} m<0$ or $\int_{\partial {\Omega}} b<0$. In this case, $\lambda_b$ is achieved and:
\begin{enumerate}
\item $\beta_1(\lambda)>0$ if $\lambda \in (0,\lambda_b)$. Moreover, if $\int_{\partial {\Omega}} b<0$ then $\beta_1(0)>0$.
\item $\lambda_b=\max\{ \lambda>0; \ \beta_1(\lambda)\geq 0\}$.
\end{enumerate} 
\item  Assume $\int_{\Omega} m<0$. Then $\lambda_b>\lambda_1(m)$ if and only if $\int_{\partial {\Omega}} b\varphi_1^q<0$. 
\item $\lambda_a>0$ if and only if either $\int_{\Omega} m<0$ or $\int_{{\Omega}} a<0$. In this case, $\lambda_a$ is achieved and:
\begin{enumerate}
\item $\alpha_1(\lambda)>0$ if $\lambda \in (0,\lambda_a)$. Moreover, if $\int_{{\Omega}} a<0$ then $\alpha_1(0)>0$.
\item $\lambda_a=\max\{ \lambda>0; \ \alpha_1(\lambda)\geq 0\}$.
\end{enumerate}
\item  Assume $\int_{\Omega} m<0$. Then $\lambda_a>\lambda_1(m)$ if and only if $\int_{{\Omega}} a\varphi_1^p<0$. 
\end{enumerate}
\end{prop}


\begin{prop}
\label{p2}
Assume $\int_{\partial {\Omega}} b<0$.
Then for every $\lambda_* \in (0,\lambda_b)$ there exist two constants $C_0=C_0(m,b), D_0=D_0(m,b)>0$ such that:
\begin{enumerate}
\item $E_{\lambda}(u)\geq C_0 \|u\|^2$  for every $\lambda \in (0,\lambda_*)$ and $u \in B_0^+$.
\item $B(u)\leq -D_0 \|u\|^q$ for every $\lambda \in (0,\lambda_*)$ and $u \in E_{\lambda,0}^-$.
\end{enumerate}
\end{prop}

The proof of Proposition \ref{p2} can be easily adapted to obtain a similar result on $a$:

\begin{prop}
\label{p2'}
 Assume $\int_{{\Omega}} a<0$.
Then for every $\lambda_* \in (0,\lambda_a)$ there exist two constants $C_0=C_0(m,a), D_0=D_0(m,a)>0$ such that:
\begin{enumerate}
\item $E_{\lambda}(u)\geq C_0 \|u\|^2$  for every $\lambda \in (0,\lambda_*)$ and $u \in A_0^+$.
\item $A(u)\leq -D_0	\|u\|^p$ for every $\lambda \in (0,\lambda_*)$ and $u \in E_{\lambda,0}^-$.
\end{enumerate}
\end{prop}

Propositions \ref{p2} and \ref{p2'} provide some kind of uniform coercivity (with respect to $\lambda$) to $E_\lambda$ on $A_0^+$ and $B_0^+$, respectively. In case $\lambda_1(m)>0$, we have the following:


\begin{prop}
\label{p2''}
Assume $\int_{\Omega} m<0$.
\begin{enumerate}
\item Given any $\mu<\lambda_1(m)$ there exists a constant $C_\mu>0$ such that $E_\lambda(u)\geq  \lambda C_\mu\|u\|^2$ for every $u \in H^1(\Omega)$ and every $\lambda \in (0,\mu)$.
\item If  $0<\lambda<\lambda_b$ then there exists $C_\lambda>0$ such that $E_\lambda(u)\geq C_\lambda \|u\|^2$ for every $u \in B_0^+$.
\item If $0<\lambda<\lambda_a$ then there exists $C_\lambda>0$ such that $E_\lambda(u)\geq C_\lambda \|u\|^2$ for every $u \in A_0^+$.
\end{enumerate}
\end{prop}

\begin{proof}
\strut
\begin{enumerate}
\item Assume by contradiction that there exist $\mu<\lambda_1(m)$ and two sequences $(\lambda_n) \subset (0,\mu)$ and $(v_n) \subset H^1(\Omega)$ such that
$$E_{\lambda_n}(v_n)<\frac{\lambda_n}{n}\|v_n\|^2.$$
Setting $w_n=\frac{v_n}{\|v_n\|}$, we may assume that $w_n \rightharpoonup w_0$ in $H^1(\Omega)$ and $\lambda_n \to \lambda_* \in [0,\mu]$. Hence $E_{\lambda_*}(w_0)\leq \limsup E_{\lambda_n}(w_n)\leq 0$. If $\lambda_*>0$ then $w_0\equiv 0$ and $w_n \rightarrow 0$, which is impossible. If $\lambda_*=0$ then $w_0$ is a constant. From
$$-\lambda_n \int_{\Omega} mw_n^2<\frac{\lambda_n}{n}$$ we get $\int_{\Omega} mw_0^2\geq 0$, which contradicts $\int_{\Omega} m<0$.\\
\item Assume that $(u_n) \subset B_0^+$ is such that
$$E_\lambda(u_n)<\frac{\|u_n\|^2}{n}.$$
Then $v_n=\frac{u_n}{\|u_n\|}$ is such that, up to a subsequence,
$v_n \rightharpoonup v_0$ in $H^1(\Omega)$, $B(v_n) \rightarrow B(v_0)$, and $E_\lambda(v_n) \leq \frac{1}{n}$. Then $B(v_0)\geq 0$ and $E_\lambda(v_0)\leq 0$. Finally, $v_0 \not \equiv 0$, since otherwise we would have $v_n \rightarrow 0$, which is impossible.
Therefore we get $\beta_1(\lambda)\leq 0$, which contradicts $\lambda<\lambda_b$.\\
\item The proof is similar to the previous one, so we omit it.
\end{enumerate}
\end{proof}







\section{Minimization in $\mathcal{N}_\lambda^+$}
\label{sec3}
\medskip 
\subsection{Minimization in $\mathcal{N}_\lambda^+ \cap B^+$}
\strut
\medskip

We set
\begin{equation}
\label{eov}
\overline{\lambda}=\begin{cases} \min\{\lambda_s,\lambda_1\} & \text{ if }  \eqref{n1} \text{ holds},\\
\min\{\lambda_s,\lambda_b\} & \text{ if } \int_{\partial \Omega} b<0 \text{ or } \int_\Omega m<0 \text{ and } \int_\Omega a<0. \end{cases}
\end{equation}

\begin{rem}
\label{eb}{\rm 
Note that if $0<\lambda <\overline{\lambda}$ then either $0<\lambda<\lambda_1$ or $0<\lambda<\lambda_b$, so that  $B_0^+ \setminus \{0\} 
\subset E_{\lambda}^+$.
}\end{rem}

Let us first prove that $\mathcal{N}_\lambda^+ \cap B^+$ is non-empty and bounded for $0<\lambda<\overline{\lambda}$:

\begin{prop}
\label{cun}
 Assume $b^+ \not \equiv 0$ and either $\int_{\partial \Omega} b<0$ or \eqref{n1} or $ \int_\Omega m<0$ and $ \int_\Omega a<0$. Then $\mathcal{N}_\lambda^+  \cap B^+ \neq \emptyset$  for every $\lambda \in (0,\overline{\lambda})$. Moreover:
\begin{enumerate}
\item If \eqref{n1} holds then for every $\mu<\overline{\lambda}$ there exists a constant $K=K_\mu>0$ such that $\|u\|\leq K\|b^+\|_{\infty}^{\frac{1}{2-q}}$ for every $u \in \mathcal{N}_\lambda^+$ and every $0<\lambda< \mu$.
\item If $\int_{\partial \Omega} b<0$ then for every $\mu<\overline{\lambda}$ there exists a constant $K=K_\mu>0$ such that $\|u\|\leq K\left(\lambda \|b^+\|_{\infty}\right)^{\frac{1}{2-q}}$ for every $u \in \mathcal{N}_\lambda^+ \cap B_0^+$ and every $0<\lambda< \mu$.
\item If $\int_\Omega a<0$ and  $\int_\Omega m<0$ then for every $\lambda<\overline{\lambda}$ there exists a constant $K_\lambda$ such that $\|u\|\leq K_\lambda \|b^+\|_{\infty}^{\frac{1}{2-q}}$ for every $u \in \mathcal{N}_\lambda^+ \cap B_0^+$.
\end{enumerate}

\end{prop}

\begin{proof}
First of all, note that since $b^+ \not \equiv 0$ we have $B^+ \neq \emptyset$. Moreover, from Remark \ref{eb}, if $u \in B^+$ then $u \in E_{\lambda}^+$.
If $u \in A_0^-$ then $j_u$ has a global minimum point $t>0$ such that $tu \in \mathcal{N}_\lambda^+ \cap B^+$. On the other hand, if $u \in A^+$ then, by Proposition \ref{p1}, there exists $t>0$ such that $tu \in \mathcal{N}_\lambda^+ \cap B^+$, so that $\mathcal{N}_\lambda^+ \cap B^+ \neq \emptyset$.\\
\begin{enumerate}
\item If \eqref{n1} holds then, given $0<\mu < \overline{\lambda}$ and $u \in \mathcal{N}_\lambda^+$ with $0<\lambda<\mu$, we may apply Proposition \ref{p2''}. Thus, for some $C_\mu,D>0$, we have
$$\lambda C_\mu \|u\|^2\leq E_\lambda(u)<\lambda \frac{p-q}{p-2} B(u)\leq \lambda D\|b^+\|_{\infty}\|u\|^q,$$
and consequently $$\|u\|\leq \left( \frac{D\|b^+\|_{\infty}}{C_\mu}\right)^{\frac{1}{2-q}}.$$ \\

\item If $\int_{\partial \Omega} b<0$ then, given $0<\mu < \overline{\lambda}$ and $u \in \mathcal{N}_\lambda^+$ with $0<\lambda<\mu$, we apply now Proposition \ref{p2}, so that
$E_\lambda(u)\geq C_0 \|u\|^2$ for every $\lambda \in (0,\mu)$ and $u \in B_0^+$. Thus, for $u \in \mathcal{N}_\lambda^+ \cap B_0^+$ we have $$C_0 \|u\|^2\leq E_\lambda(u)<\lambda \frac{p-q}{p-2} B(u)\leq \lambda D\|b^+\|_{\infty}\|u\|^q,$$ and consequently $$\|u\|\leq \left( \frac{\lambda D\|b^+\|_{\infty}}{C_0}\right)^{\frac{1}{2-q}}.$$\\

\item If $\int_\Omega a<0$ and  $\int_\Omega m<0$ then, by Proposition \ref{p2''}, for every $\lambda<\lambda_b$ there exists a constant $C_\lambda>0$ such that $E_\lambda(u)\geq C_\lambda \|u\|^2$ for $u \in B_0^+$. Thus, if $u \in \mathcal{N}_\lambda^+ \cap B_0^+$ then, for some $D>0$ there holds 
$$ C_\lambda \|u\|^2\leq E_\lambda(u)<\lambda \frac{p-q}{p-2} B(u)\leq \lambda D\|b^+\|_{\infty}\|u\|^q,$$
and consequently $$\|u\|\leq \left( \frac{\lambda D \|b^+\|_{\infty}}{C_\lambda}\right)^{\frac{1}{2-q}}.$$ 
\end{enumerate}
\end{proof}

\begin{rem}
\label{rr}{\rm 
From the proof of Proposition \ref{cun} (1), we may see that if $\int_\Omega m<0$ then  for every $\mu<\lambda_1$ there exists a constant $K=K_\mu>0$ such that $\|u\|\leq K \|b^+\|_{\infty}$ for every $u \in \mathcal{N}_\lambda^+$ and every $0<\lambda< \mu$, i.e. $\mathcal{N}_\lambda^+$ is uniformly bounded for $0<\lambda< \mu$ if $\mu<\lambda_1$. 
}\end{rem}

\begin{lem}
\label{lbb}
Let $b, \tilde{b} \in L^{\infty}(\partial \Omega)$ with $b \leq \tilde{b}$.
Then $\lambda_b\geq \lambda_{\tilde{b}}$ and $\lambda_s(b)\geq \lambda_s(\tilde{b}).$
\end{lem}

\begin{proof}
Let $B(u)=\int_{\partial \Omega} b|u|^q$ and $\tilde{B}(u)=\int_{\partial \Omega} \tilde{b}|u|^q$ for $u \in H^1(\Omega)$. Then $B(u)\leq \tilde{B}(u)$ for every $u \in H^1(\Omega)$, so that $B_0^+ \subset \tilde{B}_0^+$.
It follows that $\lambda_b\geq \lambda_{\tilde{b}}$. 
Let us set \begin{equation}
S_b(u)=\int_{\Omega} mu^2+\left(C_{pq}^{-1} B(u)A(u)^{\frac{2-q}{p-2}}\right)^{\frac{p-2}{p-q}}
\end{equation}
and
\begin{equation}
S_{\tilde{b}}(u)=\int_{\Omega} mu^2+\left(C_{pq}^{-1} \tilde{B}(u)A(u)^{\frac{2-q}{p-2}}\right)^{\frac{p-2}{p-q}}.
\end{equation}
Then $S_b(u)\leq S_{\tilde{b}}(u)$ for every $u \in H^1(\Omega)$.
Note that we can write 
$$
\lambda_s(b)=\inf \left\{ \frac{\int_{\Omega} |\nabla u|^2}{S(u)};\ u \in A_0^+ \cap B_0^+,\ S(u)>0\right\},
$$
From this formula, it follows that $\lambda_s(b)\geq \lambda_s(\tilde{b})$.
\end{proof}

\begin{rem}
\label{rbb}
From Lemma \ref{lbb} it follows that if $b_n \rightarrow b$ in $L^{\infty}(\partial \Omega)$ with $b^+ \not \equiv 0$ and $\int_{\partial \Omega} b <0$ then $\overline{\lambda}(b_n)$ is bounded away from zero. Indeed, we can fix $\tilde{b} \in L^{\infty}(\partial \Omega)$ such that $\tilde{b}^+ \not \equiv 0$, $\int_{\partial \Omega} \tilde{b} <0$ and  $b_n \leq \tilde{b}$ for  $n$ sufficiently large. Hence $\lambda_{b_n} \geq \lambda_{\tilde{b}}>0$ and $\lambda_s(b_n)\geq \lambda_s(\tilde{b})>0$ for $n$ sufficiently large.
\end{rem}

\begin{prop}
\label{pmn1}
 Assume $b^+ \not \equiv 0$ and either $\int_{\partial \Omega} b<0$ or \eqref{n1} or $\int_\Omega m<0$ and $\int_{ \Omega} a<0$. Then $\displaystyle \inf_{\mathcal{N}_\lambda^+ \cap B^+} I_{\lambda}$ is achieved by some $u_{0,\lambda}\geq 0$ for $0<\lambda<\overline{\lambda}$. Moreover:
\begin{enumerate}
\item If $\int_{\partial \Omega} b<0$ then $u_{0,\lambda} \rightarrow 0$  in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^+$.
\item If $\int_\Omega m<0<\int_{\partial \Omega} b$ and $\int_\Omega a<0$ then $u_{0,\lambda} \rightarrow c_0$  in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^+$.
\item If \eqref{n1} holds then $u_{0,\lambda} \rightarrow c_1$  in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^+$.
\end{enumerate}  
\end{prop}

\begin{proof}
Let $0<\lambda<\overline{\lambda}$. By Proposition \ref{cun}
we know that $\mathcal{N}_\lambda^+  \cap B^+$ is non-empty and bounded. We pick up a sequence $(u_n) \subset \mathcal{N}_\lambda^+  \cap B^+$ such that
$$I_{\lambda} (u_n) \rightarrow \inf_{\mathcal{N}_\lambda^+ \cap B^+} I_{\lambda}.$$
Since $(u_n)$ is bounded, we may assume that $$u_n \rightharpoonup u_0 \text{ in } H^1(\Omega)  \quad \text{and} \quad B(u_n) \rightarrow B(u_0).$$ From the shape of $j_u$ it is clear that $\displaystyle \inf_{\mathcal{N}_\lambda^+  \cap B^+} I_{\lambda}<0$. Thus $I_{\lambda}(u_0)<0$, so that $u_0 \not \equiv 0$.
Since $$E_\lambda(u_n)<\lambda \frac{p-q}{p-2}B(u_n)$$  and $u_0 \in B_0^+$ we get
$$0<E_\lambda(u_0)\leq\lambda \frac{p-q}{p-2}B(u_0),$$
i.e. $u_0 \in B^+$. From $\lambda<\lambda_s$ we deduce the existence of $t_1>0$ such that $t_1 u_0 \in \mathcal{N}_\lambda^+ \cap B^+$.
We claim that $u_n \rightarrow u_0$. Indeed, if $u_n \not \rightarrow u_0$ then $$j^{(k)}_{u_0}(t)<\liminf j^{(k)}_{u_n}(t)$$ for $k=0,1,2$ and every $t>0$. In particular, there holds 
$$0=j'_{u_0}(t_1)<j'_{u_n}(t_1)$$ for $n$ sufficiently large.
Thus $t_1>1$, since $j_{u_n}$ is decreasing in $(0,1)$. Now, since $j_{u_0}$ is decreasing in $(0,t_1)$
we get 
$$I_{\lambda}(t_1 u_0)=j_{u_0}(t_1)<j_{u_0}(1)<\liminf j_{u_n}(1)=\lim I_{\lambda}(u_n)=\inf_{\mathcal{N}_\lambda^+ \cap B^+} I_{\lambda},$$
which is a contradiction, since $t_1 u_0 \in \mathcal{N}_\lambda^+ \cap B^+$.
Therefore $u_n \rightarrow u_0$ and $t_1(u_0)=1$, so
$$I_{\lambda}(u_0)=\inf_{\mathcal{N}_\lambda^+ \cap B^+} I_{\lambda}.$$
We denote $u_0$ by $u_{0,\lambda}$.\\
\begin{enumerate}
\item If $\int_{\partial \Omega} b<0$ then, by Prop. \ref{cun}, 
$$\|u_{0,\lambda}\|\leq \left(\lambda\|b^+\|_{\infty}\right)^{\frac{1}{2-q}} K \rightarrow 0$$
as $\lambda \to 0^+$.\\
\item If $\int_\Omega m<0<\int_{\partial \Omega} b$ and $\int_\Omega a<0$, let $\lambda_n \to 0^+$ and $u_n=u_{0,\lambda_n}$. By Remark \ref{rr}, $(u_n)$ is bounded and we may assume that $u_n \rightharpoonup u_0$ in $H^1(\Omega)$.
From $$E_{\lambda_n}(u_n)=\lambda_n\left(A(u_n)+B(u_n)\right)$$
we infer that $\int_{\Omega} |\nabla u_n|^2 \rightarrow 0$, so $u_n \rightarrow u_0$ and $u_0$ is a constant.
Since $\int_\Omega m<0<\int_{\partial \Omega} b$ and $\int_\Omega a<0$, $j_{u_0}$ has a unique critical point, which is a global minimum point. Thus there is an unique constant $c$ in $\mathcal{N}_\lambda^+$. By Remark \ref{cn}, we infer that $c=c_0$, where $c_0$ is the unique zero of $\varphi$.
In particular, $c_0 \in \mathcal{N}_{\lambda_n}^+ \cap B^+$ for every $n$.
Then
$$I_{\lambda_n}(u_n)\leq I_{\lambda_n}(c_0)<0,$$
and consequently
$$-\frac{1}{2}\int_{\Omega} mu_n^2-\frac{1}{p} A(u_n)-\frac{1}{q}B(u_n)\leq -\frac{c_0^2}{2} \int_{\Omega} m-\frac{c_0^p}{p}\int_{\Omega} a -\frac{c_0^q}{q}\int_{\partial \Omega}b<0.$$
It follows that $u_n \not \rightarrow 0$, i.e. $u_0$ is a positive constant. 
Finally, since $u_n$ solves $(P_\lambda)$ for $\lambda=\lambda_n$, we have
\begin{align*}
0 & = \int_\Omega \left(\nabla u_n \nabla u_0 - \lambda_n m u_n u_0\right)  - \lambda_n \int_\Omega a u_n^{p-1} u_0 - \lambda_n \int_{\partial \Omega} b u_n^{q-1} u_0 \\
& = \lambda_n \left\{ - \int_\Omega m u_n u_0  - \int_\Omega a u_n^{p-1} u_0 - \int_{\partial \Omega} b u_n^{q-1} u_0 \right\}
\end{align*}
so, letting $n \to \infty$, we get $$u_0^{2-q} \int_{\Omega} m +u_0^{p-q} \int_{\Omega} a + \int_{\partial \Omega} b=0,$$ i.e. $u_0=c_0$.
\\ 

\item If \eqref{n1} holds then we can proceed as in the previous item to deduce that $u_n \rightarrow c$, where $c$ is a constant.
Now, by Remark \ref{cn} we infer that $c=c_1$.
\end{enumerate}
\end{proof}

\begin{rem}{\rm
Let $b_n \rightarrow b$ in $L^{\infty}(\partial \Omega)$ with $b_n^+ \not \equiv 0$ for every $n$, $b \not \equiv 0$ and $b \leq 0$. Then, from \cite[Remark 2.5]{RQU}, we have $\lambda_{b_n} \to \lambda_b>0$.
Moreover, since $b_n \rightarrow b$ in $L^{\infty}(\partial \Omega)$, we have $\int_{\partial \Omega} b_n<0$ for $n$ sufficiently large and $b_n^+ \to 0$ in $L^{\infty}(\partial \Omega)$. So
we can fix $\tilde{b} \in L^{\infty}(\partial \Omega)$ such that  $\int_{\partial \Omega} \tilde{b}<0$, $\tilde{b}^+ \not \equiv 0$, and $b_n\leq \tilde{b}$ for $n$ sufficiently large. By Remark \ref{rbb}, we have  $\lambda_s(b_n)\geq \lambda_s(\tilde{b})>0$. So $\overline{\lambda}(b_n)\geq \min\{\lambda_s(\tilde{b}), \lambda_{\tilde{b}}\}$ and
$u_{0,\lambda,b_n}$ exists for $0<\lambda<\min\{\lambda_s(\tilde{b}), \lambda_{\tilde{b}}\}$ and every $n$. Moreover, from the proof of Proposition \ref{pmn1}, for some $K_\lambda>0$ we have 
$\|u_{0,\lambda,b_n}\|\leq K_\lambda \|b_n^+\|_{\infty}^{\frac{1}{2-q}}$, and consequently $u_{0,\lambda,b_n} \to 0$ in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ and $0<\lambda<\min\{\lambda_s(\tilde{b}), \lambda_{\tilde{b}}\}$.}
\end{rem}
    
\begin{lem}
\label{leqil}
 Assume $b^+ \not \equiv 0$ and $\int_{\partial \Omega} b<0$. Then, for $0<\lambda<\overline{\lambda}$, there holds
 \begin{equation}
 \label{eqil}
 I_{\lambda}(u_{0,\lambda})<-D_0\lambda^{\frac{2}{2-q}}+o(\lambda^{\frac{2}{2-q}}),
\end{equation}  
for some $D_0>0$.
\end{lem}

\begin{proof}
Recall that for $0<\lambda<\lambda_b$ we have $B^+ \subset E_{\lambda}^+$. Then there exists $C_0>0$ such that $0<E_\lambda(u)\leq C_0\|u\|^2$ for  $u \in B^+$ and $0<\lambda<\lambda_b$.

Let $u \in B^+ \cap A_0^+$.
Then $$I_\lambda(u) \leq \frac{1}{2}E_\lambda(u)-\frac{\lambda}{q}B(u)\leq \tilde{I}_\lambda(u):=\frac{C_0}{2}\|u\|^2-\frac{\lambda}{q}B(u).$$
Thus $I_\lambda(tu)\leq \tilde{I}_\lambda(tu)$ for every $t>0$.
Note that $\tilde{I}_\lambda(tu)$ has a global minimum point $t_0$ given by
$$t_0=\left(\frac{\lambda B(u)}{C_0\|u\|^2}\right)^{\frac{1}{2-q}}.$$
and $$\tilde{I}_\lambda(t_0u)=-\frac{2-q}{2q}\lambda^{\frac{2}{2-q}}B(u)\left(\frac{B(u)}{C_0\|u\|^2}\right)^{\frac{q}{2-q}}=-D_0\lambda^{\frac{2}{2-q}},$$
where $D_0=\frac{2-q}{2q}B(u)\left(\frac{B(u)}{C_0\|u\|^2}\right)^{\frac{q}{2-q}}$.
It follows that if $I_\lambda(tu)$ has a local minimum at $t_1$ then $$I_\lambda(t_1u)< -D_0\lambda^{\frac{2}{2-q}}$$
with $D_0>0$


Let now $u \in B^+ \cap A^-$. Then $$I_\lambda(u)\leq \tilde{I}_\lambda(u):=\frac{C_0}{2}\|u\|^2-\frac{\lambda}{q}B(u)-\frac{\lambda}{p}A(u)$$ and $\tilde{I}_\lambda (tu)$ has a global minimum point $t_0$ which satisfies
$$t_0C_0\|u\|^2-\lambda t_0^{q-1}B(u)-\lambda t_0^{p-1} A(u)=0.$$
Thus $$t_0C_0\|u\|^2-\lambda t_0^{q-1}B(u)<0,$$
so that $$t_0<\left(\frac{\lambda B(u)}{C_0\|u\|^2}\right)^{\frac{1}{2-q}}.$$
Hence $$\tilde{I}_\lambda (t_0u)<-D_0\lambda^{\frac{2}{2-q}}-\lambda^{\frac{p-q+2}{2-q}}\left(\frac{B(u)}{C_0\|u\|^2}\right)^{\frac{1}{2-q}}A(u)=-D_0\lambda^{\frac{2}{2-q}}+D_1\lambda^{\frac{p-q+2}{2-q}},$$
where $D_0$ is as above and $D_1=-\left(\frac{B(u)}{C_0\|u\|^2}\right)^{\frac{p}{2-q}}\frac{A(u)}{p}$. Once again, if $I_\lambda(tu)$ has a local minimum at $t_1$ then $$I_\lambda(t_1u)<-D_0\lambda^{\frac{2}{2-q}}+D_1\lambda^{\frac{p-q+2}{2-q}}.$$
Therefore we conclude that $$\inf_{\mathcal{N}_\lambda^+ \cap B^+} I_\lambda <-D_0\lambda^{\frac{2}{2-q}}+o(\lambda^{\frac{2}{2-q}})$$
for $0<\lambda<\overline{\lambda}$.
\end{proof}   

\begin{prop}
\label{pasyc}
Assume $b^+ \not \equiv 0$ and $\int_{\partial \Omega} b<0$. If $\lambda_n \to 0^+$ then, up to a subsequence, there holds $w_{0,\lambda_n}=\lambda_n^{-\frac{1}{2-q}}u_{0,\lambda_n} \rightarrow w_0$ in $H^1(\Omega)$, where $w_0$ is a nontrivial non-negative solution of 
\begin{equation}
\label{pwl}
-\Delta w = 0 \quad\mbox{in} \ \Omega, \qquad
\frac{\partial w}{\partial \mathbf{n}} = b(x)w^{q-1} \quad \mbox{on} \
\partial \Omega.
\end{equation}
\end{prop}

\begin{proof}
Let $w_n=w_{0,\lambda_n}=\lambda_n^{-\frac{1}{2-q}}u_{0,\lambda_n}$. Since $$\|u_{0,\lambda}\|\leq \left(\lambda\|b^+\|_{\infty}\right)^{\frac{1}{2-q}} K$$ it follows that $(w_n)$ is bounded in $H^1(\Omega)$. Thus, up to a subsequence, $w_n \rightharpoonup w_0$ in $H^1(\Omega)$. Furthermore,
from Lemma \ref{leqil} we have
$$I_{\lambda_n}(u_{0,\lambda_n})<-D_0\lambda_n^{\frac{2}{2-q}}++o(\lambda_n^{\frac{2}{2-q}}),$$
with $D_0>0$.
Hence
$$\frac{\lambda_n^{\frac{2}{2-q}}}{2} \int_{\Omega} |\nabla w_n|^2 -\frac{\lambda_n^{\frac{4-q}{2-q}}}{2}\int_{\Omega} mw_n^2 -\frac{\lambda_n^{\frac{2}{2-q}}}{q} B(w_n) -\frac{\lambda_n^{\frac{2+p-q}{2-q}}}{p}
A(w_n)<-D_0\lambda_n^{\frac{2}{2-q}}++o(\lambda_n^{\frac{2}{2-q}}).$$
Dividing the above inequality by $\lambda_n^{\frac{2}{2-q}}$ and letting $n \to \infty$ we get
$$\frac{1}{2}\int_{\Omega} |\nabla w_0|^2- \frac{1}{q}B(w_0)\leq -D_0<0,$$
so $w_0 \not \equiv 0$.
Taking $v=w_n-w_0$ in
\begin{equation}
\label{ewn}
\int_{\Omega} \left(\nabla w_n \nabla v -\lambda_n m(x)w_n v -\lambda_n^{\frac{p-q}{2-q}}a(x)w_n^{p-1}v\right) -\int_{\partial \Omega} b(x)w_n^{q-1}v =0 \quad \forall v \in H^1(\Omega)
\end{equation}
and letting $n \to \infty$ 
we get $\lim \int_{\Omega} \nabla w_n \nabla (w_n-w_0) =0$, so that $w_n \rightarrow w_0$ in $H^1(\Omega)$.
Finally, \eqref{ewn} also shows that $w_0$ is a solution of  \eqref{pwl}.
\end{proof}

\medskip
\subsection{Minimization in $\mathcal{N}_\lambda^+ \cap E_\lambda^-$}
\strut
\medskip
\begin{prop}
Assume $\lambda_a, \lambda_b>0$ and $\mathcal{N}_\lambda^+ \cap E_\lambda^- \neq \emptyset$. If $\displaystyle \inf_{\mathcal{N}_\lambda^+ \cap E_\lambda^-} I_{\lambda}<0$ and $0<\lambda<\min\{\lambda_a,\lambda_b\}$ then $\displaystyle \inf_{\mathcal{N}_\lambda^+ \cap E_\lambda^-} I_{\lambda}$ is achieved.
\end{prop}

\begin{proof}
First of all, from $0<\lambda<\min\{\lambda_a,\lambda_b\}$ we have $E_\lambda^- \subset A^- \cap B^-$ and by Proposition \ref{p2'}, taking $\lambda < \lambda_* <\min\{\lambda_a,\lambda_b\}$ we get a constant $D_0>0$ such that $$A(u)\leq -D_0 \|u\|^p$$ if $u \in E_\lambda^-$. Thus, from $$\lambda A(u)=E_\lambda(u)-\lambda B(u)$$ we get
$$\lambda D_0 \|u\|^p \leq -\lambda A(u)< -E_\lambda(u)< \lambda C \|m\|_\infty \|u\|^2,$$ for some $C>0$, and consequently there exists $K>0$ such that 
\begin{equation}
\label{ebe}
\|u\|\leq K^{\frac{1}{p-2}}
\end{equation}
 if $u \in \mathcal{N}_\lambda^+ \cap E_\lambda^-$.
Let now $(u_n) \subset \mathcal{N}_\lambda^+ \cap E_\lambda^-$ be such that $$I_{\lambda}(u_n) \rightarrow \inf_{\mathcal{N}_\lambda^+ \cap E_\lambda^-} I_{\lambda}.$$ Since $(u_n)$ is bounded, we may assume that $u_n \rightharpoonup u_0$ in $H^1(\Omega)$. In particular, we have $E_\lambda(u_0)\leq 0$ and
$I_{\lambda}(u_0)\leq \displaystyle \inf_{\mathcal{N}_\lambda^+ \cap E_\lambda^-} I_{\lambda}<0$, so that $u_0 \not \equiv 0$. Since $\lambda<\min\{\lambda_a,\lambda_b\}$, we have $E_{\lambda,0}^- \setminus \{0\} \subset (A^- \cap B^-) \cup \{0\}$, so 
$u_0 \in A^- \cap B^-$. Moreover, from 
$$0>I_{\lambda}(u_0)=\frac{1}{2}E_\lambda(u_0)-\frac{\lambda}{p}A(u_0) -\frac{\lambda}{q}B(u_0),$$
we infer that $E_\lambda(u_0)<0$, i.e. $u_0 \in E_\lambda^-$.
Now, as $j_{u_0}(1)=I_{\lambda}(u_0)<0$, we see, from the shape of $j_{u_0}$, that there exists $t_2>0$ such that 
$t_2 u_0 \in \mathcal{N}_\lambda^+ \cap E_\lambda^-$, i.e. $t_2$ is a global minimum point of $j_{u_0}$.
If $u_n \not \rightarrow u_0$ then 
$$j_{u_0}(t_2)\leq j_{u_0}(1)<\liminf j_{u_n}(1)=\lim I_{\lambda}(u_n)=\inf_{\mathcal{N}_\lambda^+ \cap E_\lambda^-} I_{\lambda},$$ which is a contradiction.
Therefore $u_n \rightarrow u_0$ and consequently $t_2(u_0)=1$.
Thus $u_0 \in \mathcal{N}_\lambda^+ \cap E_\lambda^-$ and $$I_{\lambda}(u_0)=\inf_{\mathcal{N}_\lambda^+ \cap E_\lambda^-} I_{\lambda}.$$
\end{proof}

\begin{prop}
$\mathcal{N}_\lambda^+ \cap E_\lambda^- \neq \emptyset$ and $\displaystyle \inf_{\mathcal{N}_\lambda^+ \cap E_\lambda^- } I_{\lambda}<0$ in the following cases:
\begin{enumerate}
\item $\int_{\Omega} m>0>\int_{\Omega} a$, $0>\int_{\partial {\Omega}} b > -K_1(m,a)$ and $\lambda>0$.

\item  $\int_{\Omega} m<0$, $\int_{\partial {\Omega}} b\varphi_1^q<0$, 
$-\left(\frac{C_{pq}}{-\int_{\partial {\Omega}} b\varphi_1^q}\right)^{\frac{p-2}{2-q}}<\int_{\Omega} a\varphi_1^p<0$
 and $\lambda >\lambda^*$, where
\begin{equation}
\label{l*}
\lambda^*=\lambda_1(m) \left[1-\left(C_{pq}^{-1}\left(-\int_{\partial {\Omega}} b\varphi_1^q\right) \left(-\int_{\Omega} a\varphi_1^p\right)^{\frac{2-q}{p-2}}\right)^{\frac{p-2}{p-q}}\right]^{-1}.
\end{equation}
\end{enumerate}
\end{prop}

\begin{proof}
Let $u \in B^- \cap E_{\lambda}^- \cap A^-$. It is clear that if $ j_u(t)<0$ for some $t>0$ then $j_u$ has a local maximum followed by a global minimum, i.e. there are $t_1<t_2$ such that $t_1 u \in \mathcal{N}_{\lambda}^- \cap E_{\lambda}^- $ and $t_2 u \in \mathcal{N}_{\lambda}^+ \cap E_{\lambda}^- $. Moreover, in this case we have $j_u(t_2)<0$, so that $\displaystyle \inf_{\mathcal{N}_\lambda^+ \cap E_\lambda^- } I_{\lambda}<0$.
Note that $j_u(t)<0$ if and only if $i_u(t)<0$. Now $t_0(u)$ given by \eqref{eqto} is a global minimum point of $i_u$ and
\begin{eqnarray*}
i_u(t_0(u))<0 &\Leftrightarrow&
\lambda (-B(u))<C_{pq} \frac{(-E_\lambda(u))^{\frac{p-q}{p-2}}}{\left(-\lambda A(u)\right)^{\frac{2-q}{p-2}}} \\ 
&\Leftrightarrow &
(-B(u))<C_{pq} \frac{\left(\int_{\Omega} mu^2-\lambda^{-1}\int_{\Omega} |\nabla u|^2 \right)^{\frac{p-q}{p-2}}}{(-A(u))^{\frac{2-q}{p-2}}}\\
&\Leftrightarrow & E_{\lambda}(u)+\lambda \left(-C_{pq}^{-1} B(u)\right)^{\frac{p-2}{p-q}}(-A(u))^{\frac{2-q}{p-q}}<0.
\end{eqnarray*}
Note that if $$\int_{\Omega} m>0>\int_{\Omega} a, \quad 0>\int_{\partial {\Omega}} b > -K_1(m,a) \quad \text{and}  \quad \lambda>0 $$
then $c \in  B^- \cap E_{\lambda}^- \cap A^-$ for any constant $c$. Moreover, 
$$E_{\lambda}(c)+\lambda \left(-C_{pq}^{-1} B(c)\right)^{\frac{p-2}{p-q}}(-A(c))^{\frac{2-q}{p-q}}= \lambda c^2\left( -\int_{\Omega} m + \left(-\frac{1}{C_{pq}}\int_{\partial {\Omega}} b\right)^{\frac{p-2}{p-q}}\left(-\int_{\Omega} a\right)^{\frac{2-q}{p-q}}\right),$$
so that $j_c(t_0(c))<0$ for $\lambda>0$.

On the other hand, if
$$\int_{\Omega} m<0, \quad \int_{\partial {\Omega}} b\varphi_1^q<0, \quad
-\left(\frac{C_{pq}}{-\int_{\partial {\Omega}} b\varphi_1^q}\right)^{\frac{p-2}{2-q}}<\int_{\Omega} a\varphi_1^p<0, \quad \text{and}  \quad \lambda>\lambda_1,$$
then
$\varphi_1 \in B^- \cap E_{\lambda}^- \cap A^-.$ Furthermore 
$$E_{\lambda}(\varphi_1)+\lambda  \left(-C_{pq}^{-1} B(\varphi_1)\right)^{\frac{p-2}{p-q}}(-A(\varphi_1))^{\frac{2-q}{p-q}}=\lambda_1-\lambda \left[1-\left(-\frac{1}{C_{pq}}\int_{\partial {\Omega}} b\varphi_1^q\right)^{\frac{p-2}{p-q}} \left(-\int_{\Omega} a\varphi_1^p\right)^{\frac{2-q}{p-q}}\right],$$
so that $j_{\varphi_1}(t_0(\varphi_1))<0$ for $\lambda>\lambda^*$.

\end{proof}


\begin{cor}
\label{c2}
 $\displaystyle \inf_{\mathcal{N}_\lambda^+ \cap E_\lambda^-} I_{\lambda}$ is achieved by some $u_{1,\lambda}\geq 0$ in the following cases:
\begin{enumerate}
\item $\int_{\Omega} m>0>\int_{\Omega} a$, $0>\int_{\partial {\Omega}} b > -K_1(m,a)$ and $0<\lambda<\min\{\lambda_a,\lambda_b\}$.

\item  $\int_{\Omega} m<0$, $\int_{\partial {\Omega}} b\varphi_1^q<0$, 
$-\left(\frac{C_{pq}}{-\int_{\partial {\Omega}} b\varphi_1^q}\right)^{\frac{p-2}{2-q}}<\int_{\Omega} a\varphi_1^p<0$
 and $\lambda^*<\lambda <\min\{\lambda_a,\lambda_b\}$.
\end{enumerate}
\end{cor}

\begin{rem}{\rm 
Let us show that the condition $\lambda^*<\min\{\lambda_a,\lambda_b\}$ assumed in Corollary \ref{c2} (2) may indeed hold when $\int_{\Omega} m<0$. To this end, let $a_0 \in L^{\infty}(\Omega)$ and $b \in  L^{\infty}(\partial \Omega)$ be such that
$\int_{\Omega} a_0\varphi_1^p<0$ and $\int_{\partial \Omega} b\varphi_1^q<0$. Then $\lambda_{a_0}>\lambda_1$ and $\lambda_b>\lambda_1$. Let us set $a_\varepsilon=\varepsilon a_0$ for $\varepsilon>0$. As one can easily see from the definition of $\lambda_a$, we have $\lambda_{a_\varepsilon}=\lambda_{a_0}$  for every $\varepsilon>0$. Furthermore, from \eqref{l*}, we see that
$\lambda^*(m,a_\varepsilon,b) \rightarrow \lambda_1$ as $\varepsilon \to 0$. Thus, for some $\varepsilon_0>0$ there holds $\lambda^*(m,a_\varepsilon,b)<\min\{\lambda_{a_\varepsilon}, \lambda_b\}$.
The same argument applies if we consider $b_\varepsilon=\varepsilon b_0$, where $b_0 \in  L^{\infty}(\partial \Omega)$ is such that $\int_{\partial \Omega} b_0\varphi_1^q<0$. Therefore $\lambda^*(m,a_{\varepsilon},b_{\varepsilon})<\min\{\lambda_a,\lambda_b\}$ if $0<\varepsilon<\varepsilon_0$, for some $\varepsilon_0$.
}\end{rem}

\begin{prop}
\label{p3}
If $\int_{\Omega} m>0>\int_{\Omega} a$ and $0>\int_{\partial {\Omega}} b > -K_1(m,a)$  then $u_{1,\lambda} \to c_2$ in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^+$. 
\end{prop}

\begin{proof}
Let $\lambda_n \to 0^+$ and $u_n=u_{1,\lambda_n}$. From \eqref{ebe} we infer that $(u_n)$ is bounded. So, up to a subsequence, we have $u_n \rightharpoonup u_0$ in $H^1(\Omega)$. From
$$0\geq \liminf I_{\lambda_n}(u_n) \geq \int_{\Omega} |\nabla u_0|^2$$ we get $u_n \rightarrow u_0$ in $H^1(\Omega)$ and $u_0$ is a non-negative constant. If $u_0=0$ then we set $v_n=\frac{u_n}{\|u_n\|}$ and assume that $v_n \rightharpoonup v_0$ in $H^1(\Omega)$. Since $v_n \in E_{\lambda_n}^-$ we have $$\int_{\Omega} |\nabla v_0|^2 \leq \liminf E_{\lambda_n} (v_n)\leq 0$$ so $v_n \rightarrow v_0$ and $v_0$ is a positive constant. Moreover, from
$$\lambda_n\left(-\frac{1}{2}\int_{\Omega} mu_n^2 -\frac{1}{p}A(u_n) -\frac{1}{q} B(u_n)\right)<0$$ we get
$$\frac{1}{q} B(v_n)\geq -\frac{1}{2}\|u_n\|^{2-q} \int_{\Omega} mv_n^2 -\frac{1}{p}\|u_n\|^{p-q} A(v_n).$$
Thus $B(v_0)\geq 0$, which combined with $v_n \in B^-$ provides $v_0 \in B_0$ and contradicts $\int_{\partial {\Omega}} b<0$. Therefore $u_0 \neq 0$, i.e. $u_0$ is a positive constant.
Since $u_n \rightarrow u_0$ and $u_n$ is a solution of $(P_{\lambda_n})$ we have
\begin{align*}
0 & = \int_\Omega \left(\nabla u_n \nabla u_0 - \lambda_n m u_n u_0\right)  - \lambda_n \int_\Omega a u_n^{p-1} u_0 - \lambda_n \int_{\partial \Omega} b u_n^{q-1} u_0 \\
& = \lambda_n \left\{ - \int_\Omega m u_n u_0  - \int_\Omega a u_n^{p-1} u_0 - \int_{\partial \Omega} b u_n^{q-1} u_0 \right\}
\end{align*}
so $$u_0^{2-q} \int_{\Omega} m +u_0^{p-q} \int_{\Omega} a + \int_{\partial \Omega} b=0,$$ i.e. $u_0$ is a positive zero of $\varphi$. Finally, from $u_n \in N_{\lambda_n}^+$, we get
$$E_{\lambda_n}(u_n)<\lambda_n \frac{p-q}{p-2}B(u_n).$$
In particular, we have $$-\lambda_n \int_\Omega mu_n^2 <\lambda_n \frac{p-q}{p-2}B(u_n),$$
so, from $u_n \rightarrow u_0$ we get
$$u_0^{2-q}\geq \frac{p-q}{p-2} \frac{\int_{\partial \Omega}b}{\left(-\int_{\Omega} m\right)}.$$ Since $u_0$ is a positive zero of $\varphi$, we get $$u_0\geq \left(-\frac{2-q}{p-q} \frac{\int_{\Omega} m}{\int_{\Omega} a}\right)^{\frac{1}{p-2}},$$ so that, by \eqref{eph}, $u_0=c_2$.
\end{proof}


\section{Minimization in $\mathcal{N}_\lambda^-$}
\label{sec4}

\begin{prop}
\label{pch}
Assume $a^+ \not \equiv 0$ and either $\int_{\Omega} a<0$ or \eqref{n1}. Then
$\mathcal{N}_\lambda^- \cap A^+ \neq \emptyset$  for every $0<\lambda  <\min\{\lambda_a, \lambda_s\}$. Moreover, for $0<\lambda  <\min\{\lambda_a, \lambda_s\}$, there holds:
\begin{enumerate}
\item $\mathcal{N}_\lambda^- \cap A^+$ is bounded away from zero, i.e. there exists $K_\lambda>0$ such that $\|u\|\geq K_\lambda$ for $u \in \mathcal{N}_\lambda^- \cap A^+$.
\item  If $(u_n) \subset \mathcal{N}_\lambda^- \cap A^+$ is a sequence such that $(I_{\lambda}(u_n))$ is bounded from above then $(u_n)$ is bounded.
\item $\displaystyle \inf_{\mathcal{N}_\lambda^- \cap A^+} I_{\lambda}>0$.
\end{enumerate}
\end{prop}

\begin{proof}
First of all, note that since $a^+ \not \equiv 0$ we have $A^+ \neq \emptyset$. Let  $u \in A^+$. If $u \in B^-$ then $j_u$ has a global maximum point $t_1>0$, so $t_1 u \in \mathcal{N}_\lambda^-$. The same conclusion holds if $u \in B_0^+$, since $\lambda<\lambda_a$ provides $u\in E_\lambda^+$, whereas $0<\lambda<\lambda_s$ yields that $j_u$ has a global maximum point, by Proposition \ref{p1}. Therefore  $\mathcal{N}_\lambda^- \cap A^+ \neq \emptyset$.
\begin{enumerate}
\item  Let us assume first $\int_{\Omega} a<0$. Given $0<\lambda < \mu<\lambda_a$ and $u \in \mathcal{N}_\lambda^- \cap A^+$, we apply Proposition \ref{p2'}. Then, for some $C_0,D>0$, we have
\begin{equation}
\label{baaaz}
C_0\|u\|^2\leq E_\lambda(u)<\lambda \frac{p-q}{2-q} A(u)\leq \lambda D\|a^+\|_{\infty}\|u\|^p,
\end{equation}
and consequently 
\begin{equation}
\label{baaz}
\|u\|\geq \left( \frac{C_0}{\lambda D\|a^+\|_{\infty}}\right)^{\frac{1}{p-2}}.
\end{equation}
Now, if \eqref{n1} holds then, since $\lambda<\lambda_a$, there is a constant $C_\lambda>0$ such that $E_\lambda(u)\geq C_\lambda \|u\|^2$ for every $u \in A^+$. Thus
$$ C_\lambda \|u\|^2\leq E_\lambda(u)<\lambda \frac{p-q}{2-q} A(u)\leq \lambda D\|a^+\|_{\infty}\|u\|^p,$$
and consequently $$\|u\|\geq \left( \frac{C_\lambda}{\lambda D\|a^+\|_{\infty}}\right)^{\frac{1}{p-2}}$$ 
if $u \in \mathcal{N}_\lambda^- \cap A^+$.\\
\item Note that
$$I_{\lambda}(u)=\frac{p-2}{2p}E_\lambda(u)-\lambda \frac{p-q}{pq}B(u)$$  if $u \in \mathcal{N}_\lambda^-$.
Hence, as $\lambda<\lambda_a$, if in addition $u \in A^+$ then 
$$I_{\lambda}(u)\geq C_\lambda \|u\|^2 - \lambda D \|u\|^q,$$ for some constants $C_\lambda,D>0$.
From the above inequality we deduce that if $(u_n) \subset \mathcal{N}_\lambda^- \cap A^+$ is such that $(I_{\lambda}(u_n))$ is bounded from above then $(u_n)$ is bounded.\\


\item If $u \in \mathcal{N}_\lambda^-$ then
\begin{equation}
\label{ej1}
I_{\lambda}(u)=\frac{p-2}{2p}E_\lambda(u)-\lambda \frac{p-q}{qp}B(u).
\end{equation}
If, in addition, $u \in A^+$ then $E_\lambda(u)\geq C_\lambda \|u\|^2$ for some constant $C_\lambda>0$, since $\lambda<\lambda_a$.
Thus, if $u \in \mathcal{N}_\lambda^- \cap A^+ \cap B_0^-$ then
$$I_{\lambda}(u)\geq D_\lambda \|u\|^2\geq \tilde{D}_\lambda>0,$$ where we used (1).
Now, if $(u_n) \subset \mathcal{N}_\lambda^- \cap A^+ \cap B^+$ then,  from $\lambda<\lambda_s$, we have $$I_{\lambda}(u_n)=j_{u_n}(1)\geq j_{u_n}(t_0(u_n))>0.$$
If, in addition $I_{\lambda}(u_n)\rightarrow 0$
then $j_{u_n}(t_0(u_n))\rightarrow 0$, and consequently either $t_0(u_n)\rightarrow 0$ or $i_{u_n}(t_0(u_n)) \rightarrow 0$.
In the first case, we get $E_\lambda(u_n)\rightarrow 0$, so that  $u_n \rightarrow 0$ in $H^1(\Omega)$, which contradicts (1).
Now, if $i_{u_n}(t_0(u_n)) \rightarrow 0$ then
$$-\lambda B(u_n)+C_{pq} \frac{E_\lambda(u_n)^{\frac{p-q}{p-2}}}{\left(\lambda A(u_n)\right)^{\frac{2-q}{p-2}}} \rightarrow 0^+.$$ 
Since $(u_n)$ is bounded, we may assume that $$u_n \rightharpoonup u_0 \text{ in } H^1(\Omega), \quad A(u_0)=\lim A(u_n)\geq 0 \quad \text{and} \quad B(u_0)=\lim B(u_n).$$
One may easily see that if either $B(u_0)=0$ or $A(u_0)=0$ then $E_\lambda(u_n) \rightarrow 0$ and  we infer again that $u_n \rightarrow 0$ in $H^1(\Omega)$, which contradicts (1).
Thus $u_0 \in A^+ \cap B^+ \cap E_\lambda^+$. From $\lambda<\lambda_s$ we have $$0<i_{u_0}(t_0(u_0))\leq \lim i_{u_n}(t_0(u_n)),$$ which is a contradiction.
Therefore we can't have $I_{\lambda}(u_n) \rightarrow 0$, so  $\displaystyle \inf_{\mathcal{N}_\lambda^- \cap A^+} I_{\lambda}>0$.
 \end{enumerate}
\end{proof}


\begin{rem}
\label{rr2}
\strut
{\rm 
\begin{enumerate}
\item If \eqref{n1} holds and $\lambda<\lambda_1$ then the conclusions of Proposition \ref{pch} remain valid. Indeed, in this case, for every $\lambda<\lambda_1$ there exists a constant $C_\lambda>0$ such that $E_\lambda(u)\geq C_\lambda \|u\|^2$ for every $u \in H^1(\Omega)$.
\item If $\int_{\Omega} a<0$ then \eqref{baaz} shows that $\mathcal{N}_\lambda^- \cap A^+$ is uniformly bounded away from zero for $\lambda \in (0,\mu)$, with $\mu <\lambda_a$.
Furthermore, the statement in (2) can be strengthened as follows: 
if $(\lambda_n) \subset (0,\mu)$ and $(u_n) \subset  \mathcal{N}_{\lambda_n}^- \cap A^+$ are such that $I_{\lambda_n}(u_n)$ is bounded then $(u_n)$ is bounded. As a matter of fact,
in this case we have
$$I_{\lambda_n}(u_n)\geq C_0\|u_n\|^2-\mu D \|u_n\|^q$$
for some constants $C_0,D>0$.
\end{enumerate} 
}\end{rem}


\begin{prop}
\label{pmn2}
Assume $a^+ \not \equiv 0$.  If either \eqref{n1} or $\int_{\Omega} a<0$ holds  then $\displaystyle \inf_{\mathcal{N}_\lambda^- \cap A^+} I_{\lambda}$ is achieved by some $u_{2,\lambda}\geq 0$ for $0<\lambda<\min\{\lambda_s,\lambda_a\}$. Moreover, if \eqref{n1} holds then $u_{2,\lambda} \rightarrow c_2$  in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^+$.
\end{prop}

\begin{proof}
Let $0<\lambda<\min\{\lambda_s,\lambda_a\}$ and $(u_n) \subset \mathcal{N}_\lambda^- \cap A^+$ be such that
$$I_{\lambda}(u_n) \rightarrow \inf_{\mathcal{N}_\lambda^- \cap A^+} I_{\lambda}.$$
By Proposition \ref{pch}, we know that $(u_n)$ is bounded, so we may assume that $u_n \rightharpoonup u_0$ in $H^1(\Omega)$ and $u_0 \in A_0^+$. 
From 
\begin{equation}
\label{de}
\lambda \frac{p-q}{p-2}B(u_n)<E_\lambda(u_n)< \lambda \frac{p-q}{2-q}A(u_n)
\end{equation}
 we have $$E_\lambda(u_0)\leq \liminf E_\lambda(u_n) \leq \limsup E_\lambda(u_n)\leq \lambda \frac{p-q}{2-q}A(u_0).$$
If $u_0 \equiv 0$ then the above inequalities provide $E_\lambda(u_n) \rightarrow E_\lambda(u_0)$, so $u_n \rightarrow u_0\equiv 0$. This is impossible by Proposition \ref{pch} (1). Hence $u_0 \not \equiv 0$. Moreover, if $A(u_0)=0$ then $E_\lambda(u_0)\leq 0$, with $u_0 \not \equiv 0$, which contradicts $\lambda<\lambda_a$. Thus $u_0 \in A^+$ and it is easily seen that there exists $t_2>0$ such that $t_2 u_0 \in \mathcal{N}_\lambda^- \cap A^+$.
We claim that $u_n \rightarrow u_0$. Indeed, if not then 
$$j_{u_0}(t)<\liminf j_{u_n}(t)$$ for every $t>0$. 
Hence 
$$I_{\lambda} (t_2u_0)=j_{u_0}(t_2)<\liminf j_{u_n}(t_2)\leq \liminf j_{u_n}(1)=\lim I_{\lambda}(u_n)=\inf_{\mathcal{N}_\lambda^- \cap A^+} I_{\lambda}.$$
We have then a contradiction, so $u_n \rightarrow u_0$ in $H^1(\Omega)$ and
$$I_{\lambda}(u_0)=\inf_{\mathcal{N}_\lambda^- \cap A^+} I_{\lambda}.$$
We denote $u_0$ by $u_{2,\lambda}$.

Assume now \eqref{n1}. Let $\lambda_n \to 0^+$ and $u_n=u_{2,\lambda_n}$. We claim that $(u_n)$ is bounded. Indeed, assume that $\|u_n\|\rightarrow \infty$ and set $v_n=\frac{u_n}{\|u_n\|}$. We may assume that $v_n \rightharpoonup v_0$ in $H^1(\Omega)$. Note that
since $\int_\Omega a>0$ and $0<\lambda<\lambda_s$, there is a unique positive constant $c \in \mathcal{N}_{\lambda_n}^- \cap A^+$ for every $n$. Thus
$$\frac{p-2}{2p}\lambda_n A(u_n)-\frac{2-q}{2q}\lambda_n B(u_n)=I_{\lambda_n}(u_n)\leq \lambda_n \left(-\frac{\mathcal{C}^2}{2} \int_{\Omega} m-\frac{\mathcal{C}^p}{p}\int_{\Omega} a -\frac{\mathcal{C}^q}{q}\int_{\partial \Omega}b\right).$$
It follows that $I_{\lambda_n}(u_n)\rightarrow0$ and $A(v_0)=\lim A(v_n)=0$.
If $u_n \in B_0^-$ then $$I_{\lambda_n}(u_n)\geq \frac{p-2}{2p} E_{\lambda_n}(u_n),$$
so that $E_{\lambda_n}(v_n)\rightarrow 0$, and consequently $\int_{\Omega} |\nabla v_n|^2 \rightarrow 0$, i.e. $v_0$ is a nonzero constant. This contradicts $A(v_0)=0$.
On the other hand, if $u_n \in B^+$ then $u_n \in B^+ \cap A^+$ and $j_{u_n}(t_0(u_n)) \to 0$. So either $t_0(u_n) \rightarrow 0$ or 
$$-\lambda_n B(v_n)+C_{pq} \frac{E_{\lambda_n}(v_n)^{\frac{p-q}{p-2}}}{\left(\lambda_n A(v_n)\right)^{\frac{2-q}{p-2}}} \rightarrow 0^+.$$ 
Since 
\begin{eqnarray*}
t_0(u_n)&=&\left(\frac{p(2-q)}{2(p-q)} \frac{E_{\lambda_n}(u_n)}{\lambda_n A(u_n)}\right)^{\frac{1}{p-2}}=\left(\frac{p(2-q)}{2(p-q)} \left(1+\frac{B(u_n)}{A(u_n)}\right)\right)^{\frac{1}{p-2}}\\&>&\left(\frac{p(2-q)}{2(p-q)}\right)^{\frac{1}{p-2}},
\end{eqnarray*}
 the first case is ruled out.
In the second case, it follows that $E_{\lambda_n}(v_n)\rightarrow 0$ and once again we deduce that $v_0$ is a nonzero constant, which is impossible. Therefore $(u_n)$ is bounded and we may assume that $u_n \rightharpoonup u_0$ in $H^1(\Omega)$. From $$E_{\lambda_n}(u_n)=\lambda_n\left(A(u_n)+B(u_n)\right)$$
we infer that $\int_{\Omega} |\nabla u_n|^2 \rightarrow 0$, so $u_n \rightarrow u_0$ and $u_0$ is a constant.
Since $\|u_n\|\geq K>0$, we know that $u_0 \neq 0$. Finally, proceeding as in the proof of Proposition \ref{p3}, we see that $u_0=c_2$.
\end{proof}

\begin{rem}{\rm
Let $a_n \rightarrow a$ in $L^{\infty}(\Omega)$ with $a_n^+ \not \equiv 0$ for every $n$, $a \not \equiv 0$ and $a \leq 0$. Arguing as in the proof of Lemma \ref{lbb}, we may show that if $\tilde{a} \in L^{\infty}(\Omega)$ is such that $\int_\Omega \tilde{a} <0$ and $\tilde{a} \geq a_n$ for every $n$, then $\lambda_{a_n} \geq \lambda_{\tilde{a}}$ and $\lambda_s(a_n) \geq \lambda_s(\tilde{a})$ for every $n$. So $u_{2,\lambda,a_n}$ exists for $0<\lambda<\min\{\lambda_s(\tilde{a}), \lambda_{\tilde{a}}\}$ and every $n$. Moreover, since $a_n^+ \rightarrow 0$ in $L^{\infty}(\Omega)$, from \eqref{baaz} we have $\|u_{2,\lambda,a_n}\|\to \infty$  for $0<\lambda<\min\{\lambda_s(\tilde{a}), \lambda_{\tilde{a}}\}$. Finally, getting back to \eqref{baaaz}, we deduce that $\int_\Omega a_n u_{2,\lambda,a_n}^p \rightarrow \infty$ and therefore $\|u_{2,\lambda,a_n}\|_{\mathcal{C}(\overline{\Omega})}\to \infty$ for $0<\lambda<\min\{\lambda_s(\tilde{a}), \lambda_{\tilde{a}}\}$. }
\end{rem}


\begin{prop}
\label{pewi}
Assume $a^+ \not \equiv 0$ and $\int_{\Omega} a<0$. If $\lambda_n \to 0^+$ then, up to a subsequence, there holds
$w_n:=\lambda_n^{\frac{1}{p-2}}u_{2,\lambda_n} \rightarrow w_\infty$ in $H^1(\Omega)$, where $w_\infty$ is a nontrivial non-negative solution of the  problem
\begin{equation}
\label{eqwi}
-\Delta w = a(x)w^{p-1} \quad\mbox{in} \ \Omega, \qquad
\frac{\partial w}{\partial \mathbf{n}} = 0 \quad \mbox{on} \
\partial \Omega.
\end{equation}

\end{prop}

\begin{proof}
We claim that $(w_n)$ is bounded in $H_0^1(\Omega)$. Indeed, note that 
$w_n$ minimizes $J_{\lambda_n}$ over $\mathcal{M}_{\lambda_n}^- \cap A^+$, where 
$$J_\lambda(w)=\frac{1}{2}E_\lambda(w) - \frac{1}{q}\lambda^{\frac{p-q}{p-2}} B(w)-\frac{1}{p}A(w) \quad \text{for } w\in H^1(\Omega),$$
and $\mathcal{M}_\lambda$ is the Nehari manifold associated to $J_\lambda$. If $w \in \mathcal{M}_\lambda^- \cap A^+$ and $0<\lambda<\min\{\lambda_a,\lambda_s\}$ then
$$J_\lambda(w)=\left(\frac{1}{2}-\frac{1}{p}\right)E_\lambda(w)-\left(\frac{1}{q}-\frac{1}{p}\right)\lambda^{\frac{p-q}{p-2}}B(w) \geq C_0\|w\|^2 -C_1\|w\|^q$$ for some constants $C_0,C_1>0$. If we prove that $J_{\lambda_n}(w_n)$ is bounded from above then we deduce that $(w_n)$ is bounded. We have
$$J_{\lambda_n}(w_n)=\inf_{\mathcal{M}_{\lambda_n}^- \cap A^+} J_{\lambda_n} \leq \inf_{\mathcal{M}_{\lambda_n}^- \cap A^+ \cap H_0^1(\Omega)} J_{\lambda_n}.$$
If $w \in H_0^1(\Omega)$ then $B(w)=0$, so $J_\lambda(w)=\frac{1}{2}E_\lambda(w) -\frac{1}{p}A(w)$, and it can be shown that $\displaystyle \inf_{\mathcal{M}_{\lambda}^- \cap A^+ \cap H_0^1(\Omega)} J_{\lambda}$ is achieved  for $\lambda \in (0,\lambda_1^D(m))$, where $\lambda_1^D(m)$ is the first positive eigenvalue of  $$-\Delta u =\lambda m(x) u \text{ in } \Omega, \quad u=0 \text{ on }\partial \Omega.$$
Finally, we claim that the latter infimum is bounded from above for $\lambda \in (0,\lambda_1^D(m))$, which yields the conclusion.
This claim follows from the inequality
$$J_\lambda(w) \leq L(w):=\frac{C}{2}\|w\|^2 - \frac{1}{p}A(w),$$ which holds for $w \in H_0^1(\Omega)$, $\lambda \in (0,\lambda_1^D(m))$ and some $C>0$. Thus, given $w \in H_0^1(\Omega) \cap A^+$, if $J_\lambda(tw)$ achieves its global maximum at $t_0>0$ then $$J_\lambda(t_0w)\leq L(t_0w)\leq \left(\frac{1}{2}-\frac{1}{p}\right) \left(\frac{C\|w\|^2}{A(w)}\right)^{\frac{1}{p-2}}C\|w\|^2.$$
Therefore, fixing a $w_0 \in H_0^1(\Omega) \cap A^+$, we obtain
$$\inf_{\mathcal{M}_{\lambda}^- \cap A^+ \cap H_0^1(\Omega)} J_{\lambda}\leq K:=\left(\frac{1}{2}-\frac{1}{p}\right) \left(\frac{C\|w_0\|^2}{A(w_0)}\right)^{\frac{1}{p-2}}C\|w_0\|^2,$$ for $\lambda \in (0,\lambda_1^D(m))$, as claimed.
Thus $(w_n)$ is bounded in $H^1(\Omega)$, so up to a subsequence we have $w_n \rightharpoonup w_\infty$ in $H^1(\Omega)$. Taking $v=w_n-w_\infty$ in
\begin{equation}
\label{ewn2}
\int_{\Omega} \left(\nabla w_n \nabla v -\lambda_n m(x)w_n v -a(x)w_n^{p-1}v\right) -\lambda_n^{\frac{p-q}{p-2}}\int_{\partial \Omega} b(x)w_n^{q-1}v =0 \quad \forall v \in H^1(\Omega)
\end{equation}
and letting $n \to \infty$ 
we get $\lim \int_{\Omega} \nabla w_n \nabla (w_n-w_\infty) =0$, so that $w_n \rightarrow w_\infty$ in $H^1(\Omega)$. Furthermore,
since $$C_0\|w_n\|^2\leq E_{\lambda_n}(w_n)<\frac{p-q}{2-q}A(w_n)\leq C_1\|w_n\|^p$$ for some $C_0,C_1>0$, we get $\|w_n\|\geq C^{\frac{1}{p-2}}$ for some $C>0$, so that $w_\infty \not \equiv 0$.
Finally, \eqref{ewn2} also shows that $w_\infty$ is a solution of  \eqref{eqwi}.
\end{proof}

\bigskip


\section{Proofs of the main results}
\label{sec5}

Before proceeding to the proofs of our main results, we prove a partial positivity result on the boundary for nontrivial non-negative solutions of $(P_\lambda)$:

\begin{prop} \label{prop:positive}
\strut
\begin{enumerate}
\item Let $u_\lambda$ be a nontrivial non-negative solution of $(P_\lambda)$ for $\lambda > 0$. Then the set $\{ x \in \partial \Omega : u(x) = 0 \}$ has no interior points in the relative topology of $\partial \Omega$, and it is contained in $\{ x \in \partial \Omega : b(x) \leq 0 \}$ if $b\in \mathcal{C}(\partial \Omega)$.\\ 
\item Let $w_0$ be a nontrivial non-negative solution of \eqref{w0}. Then $w_0 > 0$ in $\Omega$, the set $\{ x\in \partial \Omega : w_0 = 0 \}$ has no interior points in the relative topology of $\partial \Omega$, and it is contained in $\{ x \in \partial \Omega : b(x) \leq 0 \}$ if $b\in \mathcal{C}(\partial \Omega)$
\end{enumerate}
\end{prop}

\begin{proof}
\strut
\begin{enumerate}
\item  Assume by contradiction that $x_0$ is an interior point of $\partial \Omega$ with $u_\lambda (x_0)=0$. Then, there exists $\rho_0 > 0$ such that $u_\lambda (x)=0$ for $x \in \Gamma_1 :=  B_{\rho_0} (x_0) \cap \partial \Omega$. Let $D$ be a subdomain of $\Omega$ with smooth boundary $\partial D$ such that $\Gamma_1 \subset \partial D$ and $\Gamma_0 := \partial D \setminus \overline{\Gamma_1} = \partial D \cap \Omega$. Consider the following mixed problem 
\begin{align} \label{p.v}
\begin{cases}
-\Delta u = \lambda (-m_\infty u  - a_\infty u^{p-1}) & \mbox{in $D$}, \\ 
\frac{\partial u}{\partial \mathbf{n}} = \lambda K_\infty u & \mbox{on $\Gamma_1$}, \\
u=0 & \mbox{on $\Gamma_0$}, 
\end{cases}
\end{align}
where $m_\infty = \Vert m \Vert_\infty$, $a_\infty = \Vert a \Vert_\infty > 0$, and $K_\infty > 0$ is a constant to be determined.   Arguing as in the proof of \cite[Theorem 1]{GRS09}, we can prove that if $K_\infty$ is sufficiently large then \eqref{p.v}  has a unique nontrivial non-negative weak solution $v_\lambda \in H_{\Gamma_0}^1(D)$. Here, $H_{\Gamma_0}^1(D)$ is defined as the closure of $\mathcal{C}_c^{\infty}\left(D \cup (\partial D \setminus \overline{\Gamma_0})\right)$ with respect to the $H^1(D)$ norm.
We remark that $v_\lambda \in \mathcal{C}^2(D \cup \Gamma_1) \cap \mathcal{C}(\overline{D})$ \cite{ADN,Stam}, so that $v_\lambda > 0$ in $D \cup \Gamma_1$ by the strong maximum principle and the boundary point lemma. On the other hand, 
we have $u_\lambda > 0$ on $\Gamma_0$, and for any $\varphi \in H_{\Gamma_0}^1(D)$ satisfying $\varphi \geq 0$ there holds
\begin{align*}
& \int_D \nabla u_\lambda \nabla \varphi -\lambda \int_D \left( -m_\infty u_\lambda - a_\infty u_\lambda^{p-1} \right) \varphi - \lambda \int_{\Gamma_1} K_\infty u_\lambda \varphi \geq 0, 
\end{align*}
since $u_\lambda = 0$ on $\Gamma_1$. Hence, by Proposition \ref{app:prop:comparison}, we deduce that $v_\lambda \leq u_\lambda$ in $\overline{D}$. Thus $u_\lambda (x_0) = 0 < v_\lambda (x_0)$, and a contradiction follows. 

The second assertion can be verified in a similar way. We assume that $u_\lambda (x_0)=0$ but $b(x_0) > 0$ for some $x_0 \in \partial \Omega$. Then there exist $\rho_0, b_0 > 0$ such that $b(x)\geq b_0$ for $x \in \Gamma_1 :=  B_{\rho_0} (x_0) \cap \partial \Omega$. Setting $D, \Gamma_0$ as above, we consider the following mixed problem 
\begin{align} \label{p.vv}
\begin{cases}
-\Delta u = \lambda (-m_\infty u  - a_\infty u^{p-1}) & \mbox{in $D$}, \\ 
\frac{\partial u}{\partial \mathbf{n}} = \lambda b_0 u^{q-1} & \mbox{on $\Gamma_1$}, \\
u=0 & \mbox{on $\Gamma_0$}. 
\end{cases}
\end{align}
By direct computations, we have 
\begin{align*}
\int_D \nabla u_\lambda \nabla \varphi - \lambda \int_D \left( 
-m_\infty u_\lambda - a_\infty u_\lambda^{p-1} \right) \varphi 
- \lambda \int_{\Gamma_1} b_0 u_\lambda^{q-1} \varphi \geq 0
\end{align*}
for any $\varphi\in H_{\Gamma_0}^1(D)$ satisfying $\varphi \geq 0$. Moreover, 
we have $u_\lambda > 0$ in $\Gamma_0 \cup D$, and $u_\lambda \in \mathcal{C}^\theta (\overline{D})$ for some $\theta \in (0,1)$. On the other hand, associated with \eqref{p.vv}, we consider the following eigenvalue 
problem. 
\begin{align} \label{p.e}
\begin{cases}
-\Delta \phi = \lambda (-m_\infty) \phi  + \sigma \phi & \mbox{in $D$}, \\ 
\frac{\partial \phi}{\partial \mathbf{n}} = \lambda K \phi & \mbox{on $\Gamma_1$}, \\
\phi = 0 & \mbox{on $\Gamma_0$}. 
\end{cases}
\end{align}
We note that  if  $K>0$ is sufficiently large then for every $\lambda > 0$ the above problem has a negative first eigenvalue $\sigma_1$, cf. \cite{GRS09}. Let $\phi_1$ be the positive eigenfunction associated to $\sigma_1$ with $\Vert \phi_1 \Vert_\infty = 1$. Since $\phi_1 \in \mathcal{C}^2(\Omega \cup \Gamma_1) \cap \mathcal{C}(\overline{D})$, by the strong maximum principle and the boundary point lemma, we have $\phi_1 > 0$ in $D \cup \Gamma_1$. By direct computations, if $\varepsilon > 0$ is a constant then, for any $\varphi \in H_{\Gamma_0}^1(D)$ satisfying $\varphi \geq 0$, we have 
$$
\int_D \nabla (\varepsilon \phi_1) \nabla \varphi 
- \lambda \int_D \left( -m_\infty \varepsilon \phi_1 - a_\infty (\varepsilon \phi_1)^{p-1} \right) \varphi 
- \lambda \int_{\Gamma_1} b_0 (\varepsilon \phi_1)^{q-1} \varphi \leq 0,
$$ provided that 
\begin{align*}
0< \varepsilon \leq \min \left\{ \left( \frac{-\sigma_1}{\lambda a_\infty} \right)^{\frac{1}{p-2}}, \ \left( \frac{b_0}{K} \right)^{\frac{1}{2-q}} \right\}. 
\end{align*}
Applying Proposition \ref{app:prop:comparison} to \eqref{p.vv}  with $u=\varepsilon \phi_1$ and $v=u_\lambda$, we obtain $\varepsilon \phi_1 \leq u_\lambda$ in $\overline{D}$. However, we have $u_\lambda (x_0)=0< \varepsilon \phi_1(x_0)$, which is a contradiction.\\ 

 
\item First of all, by the weak maximum principle, we have $w_0>0$ in $\Omega$. We argue now as in the previous item to deduce the positivity result on $\partial \Omega$.  As a matter of fact, it suffices to consider \eqref{p.v} replaced by the problem 
\begin{align} \label{prob:0}
\begin{cases}
-\Delta u = \lambda (w_0 u - u^2) & \mbox{in $D$}, \\
\frac{\partial u}{\partial \mathbf{n}} = 0 & \mbox{on $\Gamma_1$}, \\
u= 0 & \mbox{on $\Gamma_0$}, 
\end{cases}
\end{align}
and note that \eqref{prob:0} has a unique nontrivial non-negative weak solution for $\lambda > 0$ large. Note also that if $b$ is continuous and $b(x_0)>0=w(x_0)$ then we can apply the same argument to reach a contradiction, so that $\{ x\in \partial \Omega : w_0 = 0 \} \subset \{ x \in \partial \Omega : b(x) \leq 0 \}$. 
\end{enumerate}
\end{proof}

\begin{rem}
If $u_\lambda$ is a nontrivial non-negative solution of $(P_\lambda)$ for $\lambda < 0$ then the assertions of Proposition \ref{prop:positive} (1) hold true replacing $\{ x \in \partial \Omega : b(x) \leq 0 \}$ by $\{ x \in \partial \Omega : b(x) \geq 0 \}$. Indeed, if $\lambda < 0$ then, by the change of variables $\mu = -\lambda$, $(P_\lambda)$ reduces to 
\begin{align*}
\begin{cases}
-\Delta u= \mu \left( (-m) u + (-a)|u|^{p-2}u \right) 
& \mbox{in $\Omega$},  \\ 
\frac{\partial u}{\partial \bf{n}}= \mu (-b) |u|^{q-2}u 
& \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
with $\mu>0$.
\end{rem}
\bigskip

We deduce now our existence results using the fact that local minimizers of $I_\lambda$ constrained to $\mathcal{N} \setminus \mathcal{N}_0$ are critical points of $I_\lambda$ and, therefore, solutions of $(P_\lambda)$. It is clear that $A^+$, $B^+$ and $E_{\lambda}^-$ are open sets, so that, whenever achieved, the infima of $I_\lambda$ constrained to $\mathcal{N}_\lambda^+ \cap B^+$, $\mathcal{N}_\lambda^+ \cap E_{\lambda}^-$ and $\mathcal{N}_\lambda^- \cap A^+$ provide solutions of $(P_\lambda)$.\\


\noindent {\it Proof of Theorem \ref{t1}:}\\



By Propositions \ref{p1} and \ref{pal1}, we have $\lambda_b, \lambda_s>0$ if $\int_{\partial \Omega} b<0$ and $\lambda_a, \lambda_s>0$ if $\int_{\Omega} a<0$. Moreover, if $\int_{\partial \Omega} b<0$ then Proposition \ref{pmn1} yields that $\displaystyle \inf_{\mathcal{N}_\lambda^+ \cap B^+} I_{\lambda}$ is achieved by  $u_{0,\lambda}\geq 0$ for $0<\lambda<\min\{\lambda_b,\lambda_s\}$ and $u_{0,\lambda} \rightarrow 0$  in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^+$. Likewise, if $\int_{\Omega} a<0$ then, by Proposition \ref{pmn2}, $\displaystyle \inf_{\mathcal{N}_\lambda^- \cap A^+} I_{\lambda}$ is achieved by $u_{2,\lambda}\geq 0$ for $0<\lambda<\min\{\lambda_a,\lambda_s\}$. Furthermore, by Propositions \ref{pasyc} and \ref{pewi}, the asymptotic profiles of $u_{0,\lambda}$ and $u_{2,\lambda}$ are given by $\lambda^{\frac{1}{q-2}} w_0$ and $\lambda^{\frac{1}{p-2}} w_\infty$ as $\lambda \to 0^+$, where $w_0$ and $w_\infty$  are nontrivial non-negative solutions of \eqref{pwl} and \eqref{eqwi}, respectively.


By a standard bootstrap argument, we obtain $w_\infty \in W^{2,r}(\Omega)$, with $r>N$. The strong maximum principle and boundary point lemma yield $w_\infty > 0$ in $\overline{\Omega}$. Setting $w_\lambda = \lambda^{\frac{1}{p-2}} u_{2,\lambda}$, we have that $w_\lambda$ is bounded in $H^1(\Omega)$ and $w_\lambda$ is a weak solution of the problem
$$
\begin{cases}
-\Delta w = \lambda m w + a w^{p-1} & \text{in} \ \ \Omega, \\ 
\frac{\partial w}{\partial \mathbf{n}} = \lambda^{\frac{p-q}{p-2}}b w^{q-1} & \text{on} \ \ \partial \Omega. 
\end{cases}
$$
Rossi's bootstrap argument \cite{R} yields that $w_\lambda$ is bounded in $\mathcal{C}^\nu (\overline{\Omega})$ for some $\nu \in (0,1)$. By the compact embedding $\mathcal{C}^\nu (\overline{\Omega}) \subset \mathcal{C}^{\theta}(\overline{\Omega})$, $\theta<\nu$, we may obtain that $w_\lambda$ converges to some $w^*$ in $\mathcal{C}^{\theta}(\overline{\Omega})$. Since $w_\lambda \to w_\infty$ in $L^2(\Omega)$, we have $w^* = w_\infty$. From $w_\infty > 0$ in $\overline{\Omega}$, we have $u_{2,\lambda} > 0$ in $\overline{\Omega}$ for $\lambda > 0$ close to $0$, and $\min_{\overline{\Omega}}u_{2,\lambda} \to \infty$ as $\lambda \to 0^+$. It can be verified in a similar way that $\lambda^{-\frac{1}{2-q}}u_{0,\lambda} \to w_0$ in $C^{\tilde{\theta}}(\overline{\Omega})$ for some $\tilde{\theta}\in (0,1)$. Indeed, it suffices to note that $\lambda^{-\frac{1}{2-q}}u_{0,\lambda}$ is a weak solution of the problem
\begin{align*}
\begin{cases}
-\Delta w = \lambda m w + \lambda^{\frac{p-q}{2-q}} a w^{p-1} & \text{in} \ \ \Omega, \\ 
\frac{\partial w}{\partial \mathbf{n}} = b w^{q-1} & \text{on} \ \ \partial \Omega. 
\end{cases}
\end{align*}
Finally, Proposition \ref{prop:positive} provides the positivity properties of $w_0$.\qed \\
























\noindent {\it Proof of Theorem \ref{t1'}:}\\

By Corollary \ref{c2},  $\displaystyle \inf_{\mathcal{N}_\lambda^+ \cap E_\lambda^-} I_{\lambda}$ is achieved by $u_{1,\lambda}\geq 0$ for $0<\lambda<\min\{\lambda_a,\lambda_b\}$. Moreover, by Proposition \ref{p3}, $u_{1,\lambda} \to c_2$ in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^+$. 

Now, if $\lambda<0$ then we change the signs of  $\lambda$, $m$, $a$ and $b$. Since
$$\int_{\Omega} (-m)<0<\int_{\Omega} (-a) \quad \text{and} \quad 0<\int_{\partial \Omega} (-b)<K_1(m,a),$$
Propositions \ref{pmn1} and \ref{pmn2} yield the existence of two non-negative solutions $u_{0,-\lambda}$, $u_{2,-\lambda}$ for $0<-\lambda<\min\{\tilde{\lambda}_1, \tilde{\lambda}_s\}$, which satisfy $u_{0,-\lambda} \to c_1$ and $u_{2,-\lambda} \to c_2$ in $\mathcal{C}^{\theta}(\overline{{\Omega}})$ for some $\theta \in (0,1)$ as $\lambda \to 0^-$. We set then $v_{1,\lambda}=u_{0,-\lambda}$ and $v_{2,\lambda}=u_{2,-\lambda}$.\qed \\


\noindent {\it Proof of Theorem \ref{t3b}:}\\  

First of all, by a standard bootstrap argument we infer that $u_{2,\lambda}$ is a classical positive solution of $(P_\lambda)$, since $u_{2,\lambda} > 0$ in $\overline{\Omega}$. In order to prove that $u_{2,\lambda}$ is unstable, we consider the following linearized eigenvalue problem at $u_{2,\lambda}$ with an eigenvalue $\gamma$:
\begin{align}  \label{eigenprob}
\begin{cases}
\mathcal{L}_\lambda \psi = \lambda (p-1) a u_{2,\lambda}^{p-2}\psi + \gamma \psi & \text{in} \ \ \Omega, \\
\frac{\partial \psi}{\partial \mathbf{n}} = \lambda (q-1) b u_{2,\lambda}^{q-2} \psi + \gamma \psi & \text{on} \ \ \partial \Omega, 
\end{cases}
\end{align}
where $\mathcal{L}_\lambda = -\Delta - \lambda m$. Let us denote by $\gamma_1$ its smallest eigenvalue and by $\psi_1 \in \mathcal{C}^{2+\alpha}(\overline{\Omega})$ an eigenfunction associated to $\gamma_1$ which is positive in $\overline{\Omega}$. We claim that $\gamma_1 < 0$. To this end, we use Picone's identity \cite{BCN95}. By direct computations, we have
\begin{align*}
\int_\Omega \left( \frac{u_{2,\lambda}}{\psi_1} \right) \sum_j \frac{\partial}{\partial x_j} \psi_1^2 \frac{\partial}{\partial x_j} \left( \frac{u_{2,\lambda}}{\psi_1}\right) &= \int_\Omega \left( \frac{u_{2,\lambda}}{\psi_1} \right) (-\mathcal{L}_\lambda u_{2,\lambda}\psi_1 + u_{2,\lambda} \mathcal{L}_\lambda \psi_1) \\
& = \lambda (p-2)A(u_{2,\lambda}) + \gamma_1 \int_\Omega u_{2,\lambda}^2. 
\end{align*}
On the other hand, by Green's formula we have 
\begin{align*}
\int_\Omega \left( \frac{u_{2,\lambda}}{\psi_1} \right) \sum_j \frac{\partial}{\partial x_j} \psi_1^2 \frac{\partial}{\partial x_j} \left( \frac{u_{2,\lambda}}{\psi_1}\right) & = - \int_\Omega \psi_1^2 \left\vert \nabla \left( \frac{u_{2,\lambda}}{\psi_1} \right)\right\vert^2 + \int_{\partial \Omega} \left( \frac{u_{2,\lambda}}{\psi_1} \right) \psi_1^2 \frac{\partial}{\partial \mathbf{n}} \left( \frac{u_{2,\lambda}}{\psi_1} \right) \\
& =  - \int_\Omega \psi_1^2 \left\vert \nabla \left( \frac{u_{2,\lambda}}{\psi_1} \right)\right\vert^2 + \lambda (2-q) B(u_{2,\lambda}) - \gamma_1 \int_{\partial \Omega} u_{2,\lambda}^2. 
\end{align*}
Hence, 
\begin{align*}
\gamma_1 = \frac{- \int_\Omega \psi_1^2 \left\vert \nabla \left( \frac{u_{2,\lambda}}{\psi_1} \right)\right\vert^2 - \lambda (p-2) A(u_{2,\lambda}) + \lambda (2-q) B(u_{2,\lambda})}{\int_\Omega u_{2,\lambda}^2 + \int_{\partial \Omega} u_{2,\lambda}^2}. 
\end{align*}
Since $u_{2,\lambda} \in \mathcal{N}_\lambda$, we have $\lambda B(u_{2,\lambda}) = E_\lambda (u_{2,\lambda}) - \lambda A(u_{2,\lambda})$. So, it follows that 
\begin{align*}
\gamma_1 = \frac{- \int_\Omega \psi_1^2 \left\vert \nabla \left( \frac{u_{2,\lambda}}{\psi_1} \right)\right\vert^2 + (2-q) E_\lambda (u_{2,\lambda}) - \lambda (p-q) A(u_{2,\lambda})}{\int_\Omega u_{2,\lambda}^2 + \int_{\partial \Omega} u_{2,\lambda}^2}. 
\end{align*}
Since $u_{2,\lambda} \in \mathcal{N}_\lambda^-$, we have $E_\lambda (u_{2,\lambda})< \lambda \left( \frac{p-q}{2-q}\right) A(u_{2,\lambda})$, and hence, $\gamma_1 < 0$, as desired. The proof of Theorem \ref{t3b} is complete. \qed \\


\noindent {\it Sketch of the proof of Remark \ref{rw0}:}\\

Let us assume \eqref{assump:mbreg} and that $w_0$ is a classical positive solution of \eqref{w0}. In the same way as in the proofs of Theorems \ref{t1} and \ref{t3b},  we infer that, for $\lambda > 0$ sufficiently small, $u_{0,\lambda}$ is a classical positive solution of $(P_\lambda)$. In order to discuss the stability of $u_{0,\lambda}$, we replace $u_{2,\lambda}$ by $u_{0,\lambda}$ in \eqref{eigenprob} and analyze the sign of $\gamma_1 = \gamma_1(\lambda)$. Let $\psi_1 = \psi_1(\lambda)$ be the unique positive eigenfunction associated to $\gamma_1$ and satisfying $\int_{\partial \Omega} \psi_1^2 + \int_\Omega \psi_1^2 = 1$. Setting $\xi_\lambda (x) = \lambda m(x) + \lambda (p-1) a(x) u_{0,\lambda} (x)^{p-2}$ and $\eta_\lambda (x) = \lambda (q-1) b(x) u_{0,\lambda} (x)^{q-2}$, we observe from Theorem \ref{t1} that $\xi_\lambda \to 0$ in $L^2(\Omega)$ and $\eta_\lambda \to (q-1) b w_0^{q-2}$ in $H^1(\Omega)$ as $\lambda \to 0^+$, where $b$ is understood as an extension to $\mathcal{C}^{1+\alpha}(\overline{\Omega})$. By the continuity of $\gamma_1, \psi_1$ with respect to $\lambda$, we get to a limiting eigenvalue problem as $\lambda \to 0^+$, namely: 
\begin{align*}
\begin{cases} 
-\Delta \psi_1(0) = \gamma_1(0) \psi_1(0) & \mbox{in} \ \ \Omega, \\ 
\frac{\partial \psi_1(0)}{\partial \mathbf{n}} = (q-1)b(x) w_0^{q-2} \psi_1(0) + \gamma_1(0) \psi_1(0) & \mbox{on} \ \ \partial \Omega. 
\end{cases}
\end{align*}
By Green's formula, we have 
\begin{align} \label{eqlam0}
\int_\Omega |\nabla \psi_1(0)|^2 - \int_{\partial \Omega} (q-1)b w_0^{q-2} \psi_1(0)^2 = \gamma_1(0). 
\end{align}

Now, we claim that $\gamma_1(0) > 0$. Once this is verified, by the continuity of $\gamma_1$ we conclude that $\gamma_1(\lambda) > 0$ for $\lambda > 0$ sufficiently small, and the proof is complete. Using Green's formula again, we see that
\begin{align*}
0 = \int_\Omega (-\Delta w_0) \frac{\psi_1(0)^2}{w_0} = - \int_\Omega \left\vert \frac{\psi_1(0)}{w_0} \nabla w_0 - \nabla \psi_1(0) \right\vert^2 + \int_\Omega |\nabla \psi_1(0)|^2 - \int_{\partial \Omega}b w_0^{q-2}\psi_1(0)^2,  
\end{align*}
which combined with \eqref{eqlam0} yields
\begin{align*}
\gamma_1(0) & = (q-1) \int_\Omega \left\vert \frac{\psi_1(0)}{w_0} \nabla w_0 - \nabla \psi_1(0) \right\vert^2 + (2-q) \int_\Omega |\nabla \psi_1(0)|^2 \\
& \geq (2-q) \int_\Omega |\nabla \psi_1(0)|^2 > 0, 
\end{align*}
since $\psi_1(0)$ is not a constant. \qed \\ 





\noindent {\it Proof of Theorem \ref{t4}:}\\

Let 
\begin{align*}
\mu_{1,\pm} = \inf\left\{ \int_{D_{\pm}} \left( |\nabla u|^2 - \lambda m u^2 \right) ; \, u \in H^1_0(D_{\pm}), \, \int_{D_\pm} u^2 = 1 \right\}
\end{align*}
be the unique positive principal eigenvalues of the Dirichlet eigenvalue 
problems 
\begin{align*}
\begin{cases}
-\Delta u = \lambda m u + \mu_{\pm} u & \mbox{in} \ D_{\pm}, \\
u = 0 & \mbox{on} \ \partial D_{\pm}, 
\end{cases}
\end{align*}
and let $u_{1,\pm}$ denote the corresponding positive eigenfunctions in $H^1_0(D_{\pm})$, respectively. By a standard regularity argument and the strong maximum principle, it follows that $u_{1,\pm} \in W^{2,r}(D_{\pm})$ for any $r>N$, and $u_{1,\pm} > 0$ in $D_{\pm}$. Then, by Green's formula, we deduce 
\begin{align*}
\int_{D_{\pm}} \nabla u_{1,\pm} \nabla v - \int_{\partial D_{\pm}} \frac{\partial u_{1,\pm}}{\partial \mathbf{n}} v = \lambda \int_{D_{\pm}} mu_{1,\pm}v + 
\mu_{1,\pm} \int_{D_{\pm}} u_{1,\pm}v \quad \mbox{for all} \ v \in 
\mathcal{C}^1(\overline{D_{\pm}}). 
\end{align*}
On the other hand, for any nontrivial non-negative solution $u \in H^1(\Omega)$ of $(P_\lambda)$, we have 
\begin{align*}
\int_\Omega \nabla u \nabla w - \lambda \int_\Omega muw - \lambda \int_\Omega au^{p-1}w - \lambda \int_{\partial \Omega}bu^{q-1}w = 0 \quad \forall \ w \in H^1(\Omega),
\end{align*}
and recall that $u \in \mathcal{C}^\alpha (\overline{\Omega}) \cap W^{2,r}_{\rm loc}(\Omega)$, $r>N$, and $u>0$ in $\Omega$. 

Now, we consider $v=u$ and $w=\widetilde{u_{1,\pm}}$, where  
\begin{align*}
\widetilde{u_{1,\pm}} = \left\{ 
\begin{array}{ll}
u_{1,\pm} & \mbox{in} \ D_{\pm}, \\
0, & \mbox{otherwise}, 
\end{array} \right. 
\end{align*}
and then observe that 
\begin{align*}
0< -\int_{\partial D_{\pm}} \frac{\partial u_{1,\pm}}{\partial \mathbf{n}}u = \mu_{1,\pm} \int_{D_{\pm}} uu_{1,\pm} - \lambda \int_{D_{\pm}} a u^{p-1}u_{1,\pm}, 
\end{align*}
since $u_{1,\pm}\in \mathcal{C}^1(\overline{D_{\pm}})$,  $u_{1,\pm}>0$ in $D_{\pm}$, and $\frac{\partial u_{\pm,1}}{\partial \mathbf{n}}< 0$ from the boundary point lemma (cf. \cite{V}). Hence, it follows that $\mu_{1,+} > 0$ if $\lambda > 0$, and also that $\mu_{1,-} > 0$ if $\lambda < 0$. Since $m$ changes sign in $D_{\pm}$, we have $\mu_{1,\pm} < 0$ for $|\lambda|>\Lambda$ if $\Lambda$ is sufficiently large. Thus, we obtain $|\lambda|\leq \Lambda$. \qed 



\bigskip 
 




\subsection{Bifurcating solutions} \label{sec6}
\strut
\medskip


In this final subsection we prove Theorems \ref{t3} and \ref{thm:a=-m:bif} by a bifurcation technique. Since this technique does not require a variational structure for $(P_\lambda)$, the next results hold under the condition 
\begin{align} 
1<q<2<p. \label{1<q<2<p} 
\end{align}
We use the usual orthogonal decomposition $L^2(\Omega) = {\rm I}\!{\rm R} \oplus V$, where $$V = \left\{ v \in L^2(\Omega) : \int_\Omega v = 0 \right\},$$ and the projection $Q : L^2(\Omega) \to V$ given by $$v = Qu = u - \frac{1}{|\Omega|}\int_\Omega u.$$ In this way we reduce the problem of finding a classical positive solution to $(P_\lambda)$ under (\ref{assump:mbreg}) to the following two problems 
	\begin{align}  \label{1stpro}
	\begin{cases}
	-\Delta v + \frac{\lambda}{|\Omega|} 
	\int_{\partial \Omega} b(x) g(t + v) 
	= \lambda Q f(x, t + v) & \mbox{in $\Omega$}, \\
	%
	\dfrac{\partial v}{\partial \mathbf{n}}= \lambda b(x) g(t + v) & 
	\mbox{on $\partial \Omega$}, 
	\end{cases} 
	\end{align}
	\begin{equation} \label{2ndpro}
	\lambda \left( \int_\Omega f(x, t + v) 
	+ \int_{\partial \Omega} b g(t + v) \right) = 0, 
	\end{equation}
where $$t=\frac{1}{|\Omega|}\int_\Omega u, \quad v = Qu = u - t,\quad 
 f(x, u) = m(x)u + a(x) u^{p-1}, \quad \text{and} \quad g(u) = u^{q-1}.$$
First, to solve (\ref{1stpro}) in the H\"older space 
$\mathcal{C}^{2+\alpha}(\overline{\Omega})$, we set 
$$
X = \left\{ v \in \mathcal{C}^{2+\alpha}(\overline{\Omega}) 
: \int_\Omega v = 0 \right\}
$$ 
and introduce the nonlinear mapping $F: {\rm I}\!{\rm R} \times {\rm I}\!{\rm R} \times 
X \to Z$ given by  
	$$
	F(\lambda, t, v) = \left( -\Delta v - \lambda Q f(x, t + v) + \frac{\lambda}{|\Omega|} \int_{\partial \Omega} b g(t + v), \; 
	\frac{\partial v}{\partial \mathbf{n}} - \lambda b g(t + v) \right), 
	$$
where 
$$
Z = \left\{ (\phi, \psi) \in \mathcal{C}^{\alpha}(\overline{\Omega}) \times 
\mathcal{C}^{1+\alpha}(\partial \Omega) : 
\int_\Omega \phi + \int_{\partial \Omega} \psi = 0 \right\}.  
$$
The Fr\'echet derivative of $F$ with respect to $v$ at $(0,c,0)$ is given by $F_v (0, c, 0)v = (-\Delta v, \frac{\partial v}{\partial \mathbf{n}})$, where $c > 0$ is a constant.  From Banach's closed graph theorem it follows that $F_v (0, c, 0)$ is a homeomorphism. By the implicit function theorem, the set $F(\lambda, t, v) = 0$ consists exactly of an unique $\mathcal{C}^\infty$ function $v = v(\lambda, t)$ in a neighbourhood of $(\lambda, t) = (0,c)$, satisfying $v(0,c) = 0$. 
\par

Now, plugging $v(\lambda, t)$ in (\ref{2ndpro}), we obtain the bifurcation equation 
$$
\lambda \left( \int_\Omega f(x, t + v(\lambda, t)) + \int_{\partial \Omega} b g(t + v(\lambda, t)) \right) = 0. 
$$
From this equation we deduce that $\lambda = 0$ corresponds to the trivial solution $(\lambda, u)=(0, d)$ with $c-\varepsilon < d < c+\varepsilon$ for some $\varepsilon > 0$. 

Hence, the study of the set of non-trivial solutions 
for $(\lambda, u)$ close to $(0, c)$ is reduced to the consideration of the equation 
	\begin{equation}  \label{Phi:def}
	\Phi (\lambda, t) 
	:= \int_\Omega f(x, t + v(\lambda, t)) + 
	\int_{\partial \Omega} b g(t + v(\lambda, t)) = 0 
	\end{equation}
for $(\lambda, t)$ close to $(0, c)$.  

Recall that under the condition that $\int_\Omega m > 0 > \int_\Omega a$ and $\int_{\partial \Omega}b < 0$, the assumption $\int_{\partial \Omega} b > - \tilde{K}_1(m,a)$ made in (\ref{assump:tildK}) is equivalent to the existence of two positive zeros $c_1 < c_2$ of $\varphi$ in (\ref{ephi}).

Theorems \ref{t3} and \ref{thm:a=-m:bif} (1) are direct consequences of the following result. 
\begin{prop} \label{prop:bif} 
Assume  \eqref{cm}, \eqref{assump:mbreg}, \eqref{assump:tildK}, and \eqref{1<q<2<p}. Then the following two assertions hold. 
\begin{enumerate}
  \item[(1)]  $(P_\lambda)$ has two classical positive solutions $U_{j, \lambda}$, $j=1,2$, for $\lambda$ close to $0$, given by 
\begin{align*}
U_{j,\lambda} = t_j (\lambda) + v (\lambda, t_j(\lambda)).
\end{align*}
Here $t_j$ is a $\mathcal{C}^1$ function of $\lambda$ such that
$\lambda \mapsto U_{j,\lambda} \in \mathcal{C}^{2+\alpha}(\overline{\Omega})$ is a $\mathcal{C}^1$ map, $t_j(0) = c_j$, and $v(0,c_j) = 0$, $j=1,2$. Moreover, $U_{1,\lambda}$ is unstable (respect. asymptotically stable),  whereas $U_{2,\lambda}$ is asymptotically stable (respect. unstable) if $\lambda > 0$ (respect. $\lambda < 0$).
  \item[(2)] If $(P_\lambda)$ has a classical positive solution $u_\lambda$ with $\lambda \not= 0$ such that $u_\lambda \rightarrow c$ in $\mathcal{C}(\overline{\Omega})$ as $\lambda \to 0$, where $c>0$ is a constant, then  $\varphi (c) = 0$. 
\end{enumerate}
\end{prop}
\begin{proof} 
\strut
\begin{enumerate}
\item Differentiating $\Phi$ in \eqref{Phi:def} with respect to $t$ we find
\begin{align*}
\Phi_t (\lambda, t) 
= \int_\Omega f_u(x, t + v) (1 + v_t) 
+ \int_{\partial \Omega} b g'(t + v) (1 + v_t).  
\end{align*}
From (\ref{1stpro}) we have
\begin{align*}
\begin{cases}
-\Delta v_t + \frac{\lambda}{|\Omega|} \int_{\partial \Omega} b g'(t 
+ v) (1 + v_t) = \lambda Q [f_u (x, t + v) (1 + v_t)] & 
\mbox{in $\Omega$}, \\
\dfrac{\partial v_t}{\partial \mathbf{n}} = \lambda b g'(t + v) (1 + v_t) 
& \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
Putting $\lambda = 0$ and $t = c_j$ we get
\begin{align*}
\begin{cases}
-\Delta v_t (0,c_j)  = 0 & \mbox{in $\Omega$} \\
\dfrac{\partial v_t (0, c_j)}{\partial \mathbf{n}} = 0 & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
Since $v_t (0,c_j) \in V$, we have $v_t (0,c_j) = 0$. 
It follows that 
\begin{align*}
\Phi_t (0,c_j) 
& = \int_\Omega f_u (x, c_j) + \int_{\partial \Omega} b g'(c_j) \\
& = \int_\Omega m + (p-1)c_j^{p-2} \int_\Omega a 
+ (q-1)c_j^{q-2} \int_{\partial \Omega}b. 
\end{align*}
Therefore, 
\begin{align} \label{13082338}
\frac{c_j^{2-q}}{q-1} \Phi_t (0, c_j) 
= \frac{c_j^{2-q}}{q-1} \int_\Omega m + \left( \frac{p-1}{q-1} 
\right) c_j^{p-q} \int_\Omega a + \int_{\partial \Omega}b. 
\end{align}
Now, we claim that $\Phi_t (0, c_j) \not= 0$. Once this is verified, we end the proof of Proposition \ref{prop:bif} (1) by the use of the implicit function theorem. Let $c_0 \in (c_1, c_2)$ be the global maximum point of $\varphi$. 
This one is given explicitly by 
\begin{align} \label{c0}
c_0 = \left( \frac{(2-q)\int_\Omega m}{(p-q)(-\int_\Omega a)}\right)
^{\frac{1}{p-2}}. 
\end{align}
From (\ref{13082338}) and the fact that $\varphi (c_j)=0$, 
we deduce that 
\begin{align*} 
\frac{c^{2-q}}{q-1} \Phi_t (0,c_j) 
= \left( \frac{2-q}{q-1} \right) c_j^{2-q} \int_\Omega m 
+ \left( \frac{p-q}{q-1} \right) c_j^{p-q} \int_\Omega a. 
\end{align*}
It follows from (\ref{c0}) that 
\begin{align*}
\frac{1}{q-1} \Phi_t (0,c_j) 
&= \left( \frac{2-q}{q-1} \right) \int_\Omega m 
+ \left( \frac{p-q}{q-1} \right) c_j^{p-2} \int_\Omega a 
\\
& \left\{ \begin{array}{ll}
> \left( \frac{2-q}{q-1} \right) \int_\Omega m 
+ \left( \frac{p-q}{q-1} \right) c_0^{p-2} \int_\Omega a = 0, & \quad \text{for } j=1, \\ 
< \left( \frac{2-q}{q-1} \right) \int_\Omega m 
+ \left( \frac{p-q}{q-1} \right) c_0^{p-2} \int_\Omega a = 0, & \quad \text{for } j=2. 
\end{array} \right. 
\end{align*}
The conclusion follows. 

\bigskip 

We prove now the stability results of $U_{j,\lambda}$. We recall from \eqref{eigenprob} the linearized eigenvalue problem at $U_{j,\lambda}$:
\begin{align} \label{leigenU}
\begin{cases}
-\Delta \psi = \lambda m \psi + \lambda (p-1)U_{j,\lambda}^{p-2}\psi + \gamma \psi & \mbox{in $\Omega$}, \\
\frac{\partial \psi}{\partial \mathbf{n}}= \lambda (q-1)U_{j,\lambda}^{q-2}\psi + \gamma \psi & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align}
Let $\gamma_1 = \gamma_1(\lambda)$ be the smallest positive eigenvalue of this problem and  $\psi_1 = \psi_1(\lambda)$ be the unique positive eigenfunction associated to $\gamma_1$, satisfying $\int_\Omega \psi_1^2 + \int_{\partial \Omega} \psi_1^2 = 1$. It is easy to see that $\gamma_1(0)=0$ and $\psi_1(0)=\left( \frac{1}{|\Omega|+|\partial \Omega|}\right)^{1/2}$. 
We differentiate \eqref{leigenU} with respect to $\lambda$ and let $\lambda = 0$ to obtain
\begin{align*}
\begin{cases}
-\Delta \psi_1'(0) = m \psi_1(0) + (p-1)aU_{j,0}^{p-2}\psi_1(0) + \gamma_1'(0) \psi_1(0) & \mbox{in $\Omega$}, \\
\frac{\partial \psi_1'(0)}{\partial \mathbf{n}}= (q-1)bU_{j,0}^{q-2}\psi_1(0) + \gamma_1'(0) \psi_1(0) & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
By Green's formula, we have
\begin{align*}
& \int_\Omega \left( m \psi_1(0) + (p-1)aU_{j,0}^{p-2}\psi_1(0) + \gamma_1'(0) \psi_1(0) \right) \\ 
& \hspace*{3cm} + \int_{\partial \Omega}\left( (q-1)bU_{j,0}^{q-2}\psi_1(0) + \gamma_1'(0) \psi_1(0) \right) = 0. 
\end{align*}
Since $U_{j,0}=c_j$ and $\varphi (c_j) = 0$ from \eqref{ephi}, we deduce 
\begin{align} \label{gamm1U}
\gamma_1'(0) = -\frac{c_j^{q-2} \left\{ (2-q)c_j^{2-q}\int_\Omega m + (p-q)c_j^{p-q}\int_\Omega a \right\}}{|\Omega|+|\partial \Omega|}. 
\end{align}
By a direct computation, we see that $$c_j\varphi'(c_j)=(2-q)c_j^{2-q}\int_\Omega m + (p-q)c_j^{p-q}\int_\Omega a.$$
Since $\varphi (c_1)>0>\varphi_1(c_2)$, we deduce from \eqref{gamm1U} that $\gamma_1'(0)<0$ for $j=1$ and $\gamma_1'(0)>0$ for $j=2$, which combined with $\gamma_1(0)=0$ provides the desired conclusion. \\ 

\item Let $u$ be a classical positive solution of $(P_\lambda)$ with $\lambda \not= 0$. By Green's formula it follows that 
$$
\int_\Omega -\Delta u = - \lambda \int_{\partial \Omega} b u^{q-1}. 
$$
Hence we have 
$$
\lambda \left( \int_\Omega (mu +  a u^{p-1}) 
+ \int_{\partial \Omega} b u^{q-1} \right) = 0, 
$$
i.e.
$$
\int_\Omega (mu + a u^{p-1}) 
+ \int_{\partial \Omega} b u^{q-1} = 0. 
$$
Since $u \to c$ in $\mathcal{C}(\overline{\Omega})$, where $c$ is a positive constant, we obtain the desired conclusion. 
\end{enumerate}
\end{proof} 

Next, we recall that under the conditions $\int_\Omega m > 0 > \int_\Omega a$ and $\int_{\partial \Omega}b < 0$, the assumption $\int_{\partial \Omega} b = - \tilde{K}_1(m,a)$ is equivalent to the existence of a unique positive zero $c_0$ of $\varphi$, given by (\ref{c0}). In this case, $\varphi (c_0) = \varphi'(c_0) = 0$, and $c_0$ is the global maximum point of $\varphi$. 

\begin{lem} \label{lem:Phi}
Assume \eqref{assump:mbreg}, \eqref{1<q<2<p}, $\int_\Omega m > 0 > \int_\Omega a$, and $\int_{\partial \Omega}b = -\tilde{K}_1(m,a)$. Then, for $\Phi$ defined in \eqref{Phi:def}, we have the following:
\begin{enumerate}
\item $\Phi (0,c_0) = 0$.
\item $\Phi_t (0,c_0) = 0$.
\item $\Phi_{tt}(0,c_0) = - (2-q)(p-2) 
c_0^{-1} \int_\Omega m < 0$.
\item $\Phi_\lambda (0, c_0) = \int_\Omega mv_\lambda (0,c_0) + (p-1)c_0^{p-2} 
\int_\Omega a v_\lambda(0,c_0) 
+ (q-1)c_0^{q-2} \int_{\partial \Omega} 
b v_\lambda (0,c_0). $
\end{enumerate}
In particular, if $a=-km$ for some positive constant $k$, then 
\begin{align} \label{Philamposi}
\Phi_\lambda (0, c_0) = (q-1)c_0^{-1} \int_\Omega |\nabla v_\lambda(0,c_0)|^2 > 0. 
\end{align}
\end{lem} 
\begin{proof}
\strut
\begin{enumerate}
\item It is straightforward from
\begin{align*}
\Phi (0,c_0) &= \int_\Omega (mc_0 + a c_0^{p-1}) 
+ \int_{\partial \Omega} b c_0^{q-1} = c_0^{q-1} \varphi (c_0) = 0. 
\end{align*}\\
\item  We differentiate $\Phi$ with respect to $t$ to get
$$
 \Phi_t = \int_\Omega f_u(x, t + v)(1+v_t) + 
\int_{\partial \Omega} b g'(t + v) (1 + v_t). 
$$
Thus 
\begin{equation} \label{Phialpha0express}
\Phi_t (0,c_0) = \int_\Omega f_u(x,c_0)(1+v_t(0,c_0)) + \int_{\partial \Omega} b g'(c_0) (1 + v_t(0,c_0)). 
\end{equation}
Let us show how we derive $v_t (0,c_0)$ from (\ref{1stpro}). 
Differentiating (\ref{1stpro}) (with $v=v(\lambda, t)$) 
with respect to $t$, we obtain
\begin{align*}
\begin{cases}
-\Delta v_t + \frac{\lambda}{|\Omega|} \int_{\partial \Omega} b g'(t + v) (1 + v_t) = \lambda Q [f_u(x,t + v)(1+v_t)] & \mbox{in $\Omega$}, \\
\dfrac{\partial v_t}{\partial \mathbf{n}} 
= \lambda b g'(t + v)(1+v_t) & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
Taking $(\lambda, t)=(0,c_0)$ we get
\begin{align*} 
\begin{cases}
-\Delta v_t(0,c_0) = 0 & \mbox{in $\Omega$}, \\
\dfrac{\partial v_t(0,c_0)}{\partial \mathbf{n}} = 0 & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
Since $v_t (0,c_0) \in V$, i.e. $\int_\Omega v_t (0,c_0) = 0$, we have 
\begin{align} \label{eq140829vt0}
v_t (0,c_0)=0. 
\end{align}
Hence, from (\ref{Phialpha0express}) and $\varphi (c_0) = \varphi'(c_0)= 0$, it follows 
that 
\begin{align*}
\Phi_t (0,c_0) 
& = \int_\Omega (m + (p-1)ac_0^{p-2}) 
+ \int_{\partial \Omega} b (q-1) c_0^{q-2} \\
& = \int_\Omega (m + (p-1)a c_0^{p-2}) 
+ (q-1) c_0^{q-2} \int_{\partial \Omega} b \\
& = \int_\Omega (m + (p-1)ac_0^{p-2}) 
+ (q-1) c_0^{q-2} \left( -c_0^{p-q}\int_\Omega a - c_0^{2-q} \int_\Omega m 
\right) \\
& = (2-q) \int_\Omega m + (p-q) c_0^{p-2} \int_\Omega a = 0. 
\end{align*}\\

\item Differentiating $\Phi$ once more with respect to $t$, we have
\begin{align*}
\Phi_{tt}(0,c_0) &= \int_\Omega \{ f_{uu}(x,c_0) 
(1 + v_t (0,c_0))^2 + f_u(x,c_0) v_{tt}(0,c_0) \} \\
& \quad + \int_{\partial \Omega} b \{ g''(c_0) (1 + v_t (0,c_0))^2 
+ g'(c_0) v_{tt}(0,c_0) \}.
\end{align*}
In the same way as $v_t (0,c_0)=0$, we get $v_{tt}(0,c_0) = 0$ from 
(\ref{1stpro}). It follows that 
\begin{eqnarray*}
 \Phi_{tt}(0,c_0) 
&=& (p-1)(p-2) c_0^{p-3} \int_\Omega a + (q-1)(q-2) c_0^{q-3} 
\int_{\partial \Omega} b \\
&=& (p-1)(p-2) c_0^{p-3} \int_\Omega a + (q-1)(q-2) c_0^{q-3} 
\left( -c_0^{p-q} \int_\Omega a - c_0^{2-q} \int_\Omega m \right) \\
& =& \{ (p-1)(p-2)-(q-1)(q-2)\} c_0^{p-3}\int_\Omega a - (q-1)(q-2) c_0^{-1} 
\int_\Omega m \\
& =& c_0^{-1} \left( \{ (p-1)(p-2)-(q-1)(q-2) \} c_0^{p-2} \int_\Omega a 
- (q-1)(q-2) \int_\Omega m \right) \\
& =& c_0^{-1} \left( \{ (p-1)(p-2)-(q-1)(q-2) \} 
\frac{(2-q)\int_\Omega m}{(p-q)(-\int_\Omega a)} \int_\Omega a
- (q-1)(q-2) \int_\Omega m \right) \\
& =&c_0^{-1}\int_\Omega m \left( \frac{2-q}{p-q} \right) 
\{ (q-1)(q-2) + (p-1)(p-2) - (q-1)(p-q) \} \\
& = &c_0^{-1}\int_\Omega m\left( \frac{2-q}{p-q} \right) 
(p-2)(q-p) =
-(2-q)(p-2) c_0^{-1}\int_\Omega m  < 0,
\end{eqnarray*}
where we have used again that $\varphi(c_0)=\varphi'(c_0)=0$. \\
 

\item From the formula 
\begin{align*}
\Phi_\lambda = \int_\Omega f_u(x,t + v)v_\lambda 
+ \int_{\partial \Omega} b g'(t + v) v_\lambda,  
\end{align*}
it follows that 
\begin{align*}
\Phi_\lambda (0,c_0) 
&= \int_\Omega f_u(x, c_0)v_\lambda (0, c_0) 
+ \int_{\partial \Omega} b g'(c_0) v_\lambda (0,c_0) \nonumber \\
&= \int_\Omega (m + (p-1) a c_0^{p-2}) v_\lambda (0, c_0) 
+ \int_{\partial \Omega} b (q-1) c_0^{q-2} v_\lambda (0,c_0) \nonumber \\
&= \int_\Omega m v_\lambda (0,c_0) + (p-1) c_0^{p-2} \int_\Omega a 
v_\lambda (0,c_0) + (q-1)c_0^{q-2} \int_{\partial \Omega} 
b v_\lambda (0,c_0). 
\end{align*}
\\

\item From (\ref{1stpro}) we get 
\begin{align*}
\begin{cases}
-\Delta v_\lambda + \frac{1}{|\Omega|} \int_{\partial \Omega} b 
[g(t + v) + \lambda g'(t + v) v_\lambda] 
= Q[f(x, t + v)] + \lambda Q[f_u(x, t + v) v_\lambda] & 
\mbox{in $\Omega$}, \\
\dfrac{\partial v_\lambda}{\partial \mathbf{n}} = b [g(t + v) + \lambda  g'(t + v) v_\lambda] & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
Put $\lambda = 0$, $t = c_0$, and $v(0,c_0)=0$, to obtain
\begin{align*}
\begin{cases}
-\Delta v_\lambda (0,c_0) 
+ \frac{1}{|\Omega|} \int_{\partial \Omega} b g(c_0) 
= Q[f(x, c_0)] & \mbox{in $\Omega$}, \\ 
\dfrac{\partial v_\lambda (0,c_0)}{\partial \mathbf{n}} 
= b g(c_0) & \mbox{on $\partial \Omega$}, 
\end{cases}
\end{align*}
where $$Q[f(x,c_0)] = mc_0 + a c_0^{p-1} - \frac{1}{|\Omega|} \int_\Omega 
(mc_0 + a c_0^{p-1}).$$ It follows that 
\begin{align*}
\begin{cases}
-\Delta v_\lambda (0,c_0) 
+ \frac{1}{|\Omega|} \{ \int_\Omega (mc_0 +a c_0^{p-1}) + 
\int_{\partial \Omega} b c_0^{q-1} \} 
= mc_0 + a c_0^{p-1} & \mbox{in $\Omega$}, \\
\dfrac{\partial v_\lambda (0,c_0)}{\partial \mathbf{n}} 
= b c_0^{q-1} & \mbox{on $\partial \Omega$},  
\end{cases}
\end{align*}
and consequently 
\begin{align*}
\begin{cases}
-\Delta v_\lambda (0,c_0) 
= mc_0 +a c_0^{p-1} & \mbox{in $\Omega$}, \\ 
\dfrac{\partial v_\lambda (0,c_0)}{\partial \mathbf{n}} 
= b c_0^{q-1} & \mbox{on $\partial \Omega$},  
\end{cases}
\end{align*}
since $\varphi(c_0)=0$. Hence
\begin{align*}
\int_\Omega |\nabla v_\lambda (0,c_0)|^2 - c_0^{q-1} \int_{\partial \Omega} b 
v_\lambda (0,c_0) = \int_\Omega (mc_0 + ac_0^{p-1}) v_\lambda (0,c_0). 
\end{align*}
From (4) we get
\begin{eqnarray*}
 c_0\Phi_\lambda(0,c_0)  &=& c_0 \int_\Omega mv_\lambda(0,c_0) + (p-1)c_0^{p-1}\int_\Omega a v_\lambda(0,c_0) + (q-1)c_0^{q-1}\int_{\partial \Omega}bv_\lambda(0,c_0) \\
& =& (q-1)\int_\Omega |\nabla v_\lambda (0,c_0)|^2 + \int_\Omega \left\{ 
(2-q)c_0 m + (p-q)c_0^{p-1} a \right\} v_\lambda(0,c_0).  
\end{eqnarray*}
Since $k$ is a positive constant and $a=-km$, we have
\begin{align*}
(2-q)c_0 m + (p-q)c_0^{p-1} a &= mc_0 \left\{ (2-q) - (p-q)c_0^{p-2}k 
\right\} \\ 
& = mc_0 \left\{ 
(2-q) - (p-q) \frac{(2-q)\int_\Omega m}{(p-q)(-\int_\Omega a)}k \right\}
\\ 
& = 0. 
\end{align*}
Therefore
\begin{align*}
c_0\Phi_\lambda(0,c_0) = (q-1) \int_\Omega |\nabla v_\lambda(0,c_0)|^2. 
\end{align*}
Moreover, since $b\not\equiv 0$, $v_\lambda(0,c_0)$ is not a constant, so that $\int_\Omega |\nabla v_\lambda(0,c_0)|^2 > 0$. 
\end{enumerate} 

The proof of Lemma \ref{lem:Phi} is now complete. 
\end{proof}

Theorem \ref{thm:a=-m:bif} (2) is then a direct consequence of the following result:
\begin{prop} \label{prop:turn}
Assume  \eqref{assump:mbreg}, \eqref{1<q<2<p}, $a=-km$ for some positive constant $k$ and $$\int_\Omega m > 0 > \int_{\partial \Omega}b = -\tilde{K}_1(m,a).$$ Then there exists a constant $\varepsilon > 0$ and a $\mathcal{C}^3$ function $\lambda:(c_0-\varepsilon,c_0+\varepsilon)\rightarrow {\rm I}\!{\rm R}
$ satisfying $\lambda (c_0) = \lambda'(c_0) = 0$ and $\lambda''(c_0) > 0$, such that the set 
\begin{align*}
\{ (\lambda (t), \ t + v(\lambda (t), t)) : 
t \in (c_0-\varepsilon,c_0+\varepsilon) \}
\end{align*}
is contained in the positive solutions set of $(P_\lambda)$. Moreover, the positive solution $t + v(\lambda (t), t)$ of $(P_{\lambda (t)})$ is asymptotically stable for $c_0 < t < c_0 + \varepsilon$ and unstable for $c_0 - \varepsilon < t < c_0$. 
\end{prop}

\begin{rem}{\rm 
From \eqref{c0} note that if $a=-km$ then $c_0 = c_0(k) = \left( \frac{2-q}{(p-q)k}\right)^{\frac{1}{p-2}}$. It follows that $k \mapsto c_0(k)$ is decreasing, $\lim_{k \to 0^+} c_0(k)=\infty$ and $\lim_{k \to \infty} c_0(k)=0$.
}\end{rem} 


\begin{proof} 
By (\ref{Philamposi}) and the implicit function theorem,  we deduce that there exists a $\mathcal{C}^3$ function $t \mapsto \lambda (t)$  such that 
\begin{align*}
\Phi (\lambda, t) = 0 \ \ \mbox{for $(\lambda, t)$ close to $(0,c_0)$} \Longleftrightarrow 
(\lambda, t) = (\lambda (t), t) \ \ \mbox{for $t$ close to $c_0$}.
\end{align*}
From Lemma \ref{lem:Phi} (1) we have $\lambda (c_0) = 0$, whereas  from Lemma \ref{lem:Phi} (2), (3), and (\ref{Philamposi}), we have
\begin{align*}
\lambda'(c_0) = - \frac{\Phi_t (0,c_0)}{\Phi_\lambda (0,c_0)} = 0, 
\quad 
\lambda''(c_0) = -\frac{\Phi_{tt} (0,c_0)}{\Phi_\lambda (0,c_0)} 
>0, 
\end{align*}
as desired. 
 
We prove now the stability result. Recall $c_0$ is the unique zero of $\varphi$, given by \eqref{ephi}, as well as its global maximum point. So we have
\begin{align} \label{eq:140828c0i}
& c_0^{q-2}\varphi (c_0) =  \int_\Omega m + c_0^{p-2} \int_\Omega a + c_0^{q-2}\int_{\partial \Omega}b=0, \\
& c_0^{q-1}\varphi' (c_0)= (2-q)\int_\Omega m + (p-q)c_0^{p-2} \int_\Omega a = 0. \label{eq:140828c0ii}
\end{align} 
We let $w(t):= t + v(\lambda (t), t)$, and consider the stability of $(\lambda (t), w(t))$, $|t-c_0|<\varepsilon$. To this end, we study the linearized eigenvalue problem at $(\lambda (t), w(t))$, which is given by
\begin{align} \label{prob:eigen140829}
\begin{cases} 
-\Delta \psi = \lambda m \psi + \lambda a (p-1) w^{p-2}\psi + \gamma \psi & \mbox{in $\Omega$}, \\ 
\partial_{\mathbf{n}}\psi = \lambda b (q-1) w^{q-2}\psi + \gamma \psi & \mbox{on $\partial \Omega$}.
\end{cases}
\end{align}
Let $\gamma_1 = \gamma_1(t)$ be its smallest eigenvalue, which is simple, and $\psi_1 = \psi_1(t)$ be the positive eigenfunction associated to $\gamma_1$ satisfying $ \int_\Omega \psi_1^2 + \int_{\partial \Omega} \psi_1^2=1$. 

First, we claim that
\begin{align} \label{val140829gamm10}
\gamma_1(c_0) = 0, \quad \mbox{and} \ \ \psi_1(c_0)\equiv \left( \frac{1}{|\Omega|+|\partial \Omega|}\right)^{1/2}. 
\end{align}
Indeed, putting $t=c_0$ in \eqref{prob:eigen140829}, we have 
\begin{align*}
\begin{cases}
-\Delta \psi_1(c_0) = \gamma_1(c_0) \psi_1(c_0) & \mbox{in $\Omega$}, \\ 
\partial_{\mathbf{n}}\psi_1 = \gamma_1(c_0) \psi_1(c_0) & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
By uniqueness, it follows that $\gamma_1(c_0)=0$ and 
$$
\psi_1(c_0)\equiv \left( \frac{1}{|\Omega|+|\partial \Omega|}\right)^{1/2}, 
$$
as claimed. 

Second, we show that
\begin{align} \label{val140829gamm11}
\gamma_1'(c_0) =0, \quad \mbox{and} \ \ \psi_1'(c_0)= 0.
\end{align}
To this end, we differentiate \eqref{prob:eigen140829} with respect to $t$ (with $\gamma=\gamma_1$ and $\psi=\psi_1$) to obtain 
\begin{align} \label{prob:140828psi1pri}
\begin{cases}
-\Delta \psi_1' = m\left(\lambda' \psi_1 + \lambda \psi_1' \right) +(p-1)a\left( \lambda' w^{p-2}\psi_1 + \lambda (p-2) w^{p-3} w'\psi_1  + \lambda  w^{p-2} \psi_1'\right) \\
\hspace*{2cm} + \gamma_1' \psi_1 + \gamma_1 \psi_1' & \mbox{in $\Omega$}, \\ 
\partial_{\mathbf{n}}\psi_1' = b (q-1)\left(\lambda' w^{q-2}\psi_1 + \lambda  (q-2)w^{q-3}w' \psi_1 + \lambda  w^{q-2}\psi_1' \right)
+ \gamma_1' \psi_1 + \gamma_1 \psi_1' & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align}
Taking $t=c_0$  in \eqref{prob:140828psi1pri} we get
\begin{align*}
\begin{cases}
-\Delta \psi_1'(c_0) = \gamma_1'(c_0) \psi_1(c_0) & \mbox{in $\Omega$}, \\ 
\partial_{\mathbf{n}}\psi_1'(c_0) = \gamma_1'(c_0) \psi_1(c_0) & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
By Green's formula, we deduce that
\begin{align*}
& \int_\Omega (\gamma_1'(c_0) \psi_1(c_0)) + \int_{\partial \Omega} (\gamma_1'(c_0) \psi_1(c_0)) = 0, 
\end{align*}
so that $\gamma_1'(c_0) \psi_1(c_0) (|\Omega|+|\partial \Omega|)=0$. Hence, we have $\gamma_1'(c_0) = 0$, and consequently $\psi_1'(c_0)$ is a constant. On the other hand, differentiating $\int_\Omega \psi_1^2 + \int_{\partial \Omega} \psi_1^2 = 1$ with respect to $t$ we obtain 
\begin{align} \label{eq:140828psipsipri}
\int_\Omega \psi_1 \psi_1' + \int_{\partial \Omega} \psi_1 \psi_1' = 0. 
\end{align}
Thus we have $\psi_1(c_0) \psi_1'(c_0) (|\Omega|+|\partial \Omega|) = 0$, which implies $\psi_1'(c_0) = 0$, as desired.

Third, we verify that
\begin{align} \label{val140829gamm12}
\gamma_1''(c_0)=0.
\end{align}
Differentiating \eqref{prob:140828psi1pri} with respect to $t$, we obtain
\begin{align} \label{prob140828psi13}
\begin{cases}
-\Delta \psi_1'' = m\left( \lambda''  \psi_1 + 2\lambda'  \psi_1' + \lambda  \psi''\right) +(p-1)a\left(  \lambda'' w^{p-2}\psi_1 + \lambda' (w^{p-2}\psi_1)' \right)& \\ 
\hspace*{1.5cm} + \left[\lambda a (p-1)\left((p-2) w^{p-3} w' \psi_1 +  w^{p-2} \psi_1'\right)\right]' + \gamma_1'' \psi_1 + 2 \gamma_1' \psi_1' + \gamma_1 \psi_1'' & \mbox{in $\Omega$}, \\ 
\partial_{\mathbf{n}}\psi_1'' = b (q-1)\left[\lambda''  w^{q-2}\psi_1 + \lambda'  (w^{q-2}\psi_1)' \right]  + \left[\lambda b (q-1)\left( (q-2)w^{q-3}w' \psi_1 +  w^{q-2}\psi_1'\right)\right]' & \\
\hspace*{1.5cm} + \gamma_1'' \psi_1 + 2\gamma_1' \psi_1' + \gamma_1 \psi_1'' & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align}
Taking $t=c_0$ we get
\begin{align*}
\begin{cases}
-\Delta \psi_1''(c_0) = \lambda''(c_0) m \psi_1(c_0) + \lambda''(c_0) a (p-1) 
(w(c_0))^{p-2} \psi_1(c_0) + \gamma_1''(c_0) \psi_1(c_0) & \mbox{in $\Omega$}, \\ 
\partial_{\mathbf{n}}\psi_1''(c_0) = \lambda''(c_0) b (q-1) (w(c_0))^{q-2}\psi_1(c_0) + \gamma_1''(c_0) \psi_1(c_0) & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
Since $\psi_1(c_0)$ is a positive constant, it follows by Green's formula that 
\begin{align*}
& \lambda''(c_0)\left( 
\int_\Omega m + (p-1) \int_\Omega a(w(c_0))^{p-2} + (q-1)\int_{\partial \Omega}
b(w(c_0))^{q-2} \right) \\
& \hspace*{1.5cm}+ \gamma_1''(c_0) (|\Omega|+|\partial \Omega|) = 0.
\end{align*}
Note that $w(c_0)=c_0$, which combined with \eqref{eq:140828c0i} and \eqref{eq:140828c0ii} implies 
\begin{align} \label{eq1408281842}
\int_\Omega m + (p-1) \int_\Omega a(w(c_0))^{p-2} + (q-1)\int_{\partial \Omega}
b(w(c_0))^{q-2} = 0,
\end{align}
and \eqref{val140829gamm12} follows. 

Finally, we verify that
\begin{align} \label{140829gamm13posi} 
\gamma_1'''(c_0) > 0. 
\end{align}
We differentiate once more \eqref{prob140828psi13} with respect to $t$ and take $t=c_0$. Since $\lambda (c_0)=\lambda'(c_0)=0$ and $\psi_1'(c_0)=0$ we deduce that
\begin{align*}
\begin{cases}
-\Delta \psi_1'''(c_0) = \lambda'''(c_0)m\psi_1(c_0)  + \lambda'''(c_0)a(p-1) (w(c_0))^{p-2} \psi_1(c_0) & \\
\hspace*{2cm} + 3\lambda''(c_0)a(p-1)(p-2) (w(c_0))^{p-3}w'(c_0)\psi_1(c_0) + \gamma_1'''(c_0)\psi_1(c_0) & \mbox{in $\Omega$}, \\
\partial_{\mathbf{n}}\psi_1'''(c_0) = \lambda'''(c_0)b(q-1)(w(c_0))^{q-2}\psi_1(c_0)  + 3 \lambda''(c_0)b(q-1)(q-2)(w(c_0))^{q-3}w'(c_0)\psi_1(c_0) 
& \\
\hspace{2cm} + \gamma_1'''(c_0)\psi_1(c_0) & \mbox{on $\partial \Omega$}. 
\end{cases}
\end{align*}
Since $w(c_0)=c_0$ and $\psi_1(c_0)$ is a positive constant, by Green's formula, we deduce that
\begin{align*}
0&= \gamma_1'''(c_0) (|\Omega|+|\partial \Omega|) + \lambda'''(c_0) \left\{ \int_\Omega \left( m + (p-1)a c_0^{p-2} \right) + (q-1) c_0^{q-2} \int_{\partial \Omega}b \right\} \\
&+ 3\lambda''(c_0) \left\{ (p-1)(p-2)c_0^{p-3} \int_\Omega a w'(c_0) + (q-1)(q-2) c_0^{q-3} \int_{\partial \Omega}b w'(c_0) \right\}. 
\end{align*}
Using \eqref{eq1408281842}, we see that the second term on the right hand side vanishes. In order to deal with the third term, we consider $w'(c_0)$. Note that, from the definition of $w$, we have $w'(t) = 1 + v_\lambda \lambda' + v_t$. Since $\lambda'(c_0)=0$, we have then $w'(c_0) = 1 + v_t(0,c_0) = 1$, where we have used \eqref{eq140829vt0}. From $a=-km$, it follows that 
$$
 \gamma_1'''(c_0) (|\Omega|+|\partial \Omega|) \\
 = 3\lambda''(c_0) \left\{ k(p-1)(p-2)c_0^{p-3} \int_\Omega m + (q-1)(2-q) c_0^{q-3} \int_{\partial \Omega}b \right\}.
$$

Now, we recall the assumption
\begin{align} \label{eq140829bm} 
\int_{\partial \Omega}b = -\tilde{K}_1(m,-km),
\end{align}
which yields $\int_{\partial \Omega}b = -\tilde{C}_{pq}k^{\frac{q-2}{p-2}}\int_\Omega m$, where 
\begin{align*}
\tilde{C}_{pq} = \left( \frac{q}{2} \left( \frac{p}{2} \right)^{\frac{2-q}{p-2}} \right)^{-1} \frac{q(p-2)}{2(p-q)} \left( \frac{p(2-q)}{2(p-q)} \right)^{\frac{2-q}{p-2}} = \frac{p-2}{p-q} \left( \frac{2-q}{p-q}\right)^{\frac{2-q}{p-2}}.  
\end{align*}
Moreover, from \eqref{eq:140828c0ii} we also have
\begin{align*}
c_0^{p-q} = k^{\frac{q-p}{p-2}}\left( \frac{2-q}{p-q}\right)^{\frac{p-q}{p-2}}. \end{align*}
Thus, from \eqref{eq140829bm} we deduce that
\begin{eqnarray*}
 \gamma_1'''(c_0) (|\Omega|+|\partial \Omega|)  
& =& 3\lambda''(c_0) c_0^{q-3} \left\{ (p-1)(p-2)c_0^{p-q}\int_\Omega m + (q-1)(2-q) \int_{\partial \Omega}b \right\}  \\
& =& 3\lambda''(c_0) c_0^{q-3} \left( \int_\Omega m \right) k^{\frac{q-2}{p-2}}\left( \frac{2-q}{p-q}\right)^{\frac{2-q}{p-2}} (p-2)(2-q). 
\end{eqnarray*}
From $\lambda''(c_0)>0$ and $\int_\Omega m > 0$, we infer \eqref{140829gamm13posi}. 

Summing up, from \eqref{val140829gamm10}, \eqref{val140829gamm11}, \eqref{val140829gamm12}, and \eqref{140829gamm13posi}, the desired conclusion follows. 

The proof of Proposition \ref{prop:turn} is now complete. 
\end{proof} 

\bigskip 





