\section{Introduction and statement of  results}\label{sec1}}
\setcounter{equation}{0}
\par
In this paper,  we study the existence and asymptotic behavior of positive vector solutions for  the following coupled Kirchhoff-type system in $\R^3$:
\begin{equation}\label{11}
\begin{cases}
-\left(a_1+b_1\int_{\R^3}|\nabla u|^2dx\right)\Delta u+\lambda V(x)u=\frac{\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},&x\in\R^3,\\
-\left(a_2+b_2\int_{\R^3}|\nabla v|^2dx\right)\Delta v+\lambda W(x)v=\frac{\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,&x\in\R^3,\\
u,v\in \mathcal{D}^{1,2}(\R^3),
\end{cases}\tag{$\mathcal{K}_{\bf{b},\lambda}$}
\end{equation}
where $a_i>0$ are constants, $\lambda,b_i>0$ are  parameters for $i=1,2$, $\alpha>1$, $\beta>1$ satisfy $\alpha+\beta\le4$. We assume that $V(x)$ and $W(x)$ satisfy the following hypotheses:
\vskip4mm
\par\noindent
\begin{tabular}{@{}rp{14.80cm}}
	$(H_1)$ & $V(x),W(x)\in C(\R^3,[0,\infty))$. $\Omega_1=\textup{int } V^{-1}(0)$ and $\Omega_2=\textup{int } W^{-1}(0)$ have smooth boundaries with $\overline{\Omega}_1= V^{-1}(0)$, $\overline{\Omega}_2= W^{-1}(0)$, and ${\Omega}_1\cap\Omega_2\neq\emptyset$;\\
	$(H_2)$&There exists  $c>0$ such that the set $\mathcal{M}=\{x\in\R^3:V(x)W(x)\leq c^2\} $ has finite positive Lebesgue measure.   \\
\end{tabular}
\vskip4mm
\par
This kind of conditions, first proposed by Bartsch--Wang\cite{BW}, implies that $\lambda V$ and $\lambda W$ represent potential wells whose depths are controlled by $\lambda$, and has attracted the attention of several researchers, see \cite{BPW,SJT1,FSX,DYB,AN} and the
references therein. $\lambda V$ and $\lambda W$ are called steep potential wells if $\lambda$ is large enough. We note that condition $(H_2)$ implies that $\Omega_1$ and $\Omega_2$ have finite positive Lebesgue measures.
\par
In recent years, the following scalar Kirchhoff-type equation 
\begin{equation}\label{1}
\begin{cases}
-\left(a+b\int_{\R^3}|\nabla u|^2dx\right)\Delta u+ V(x)u=f(x,u),\hspace{1ex}x\in\R^3,\\
u\in H^1(\R^3),
\end{cases}
\end{equation}
where $V:\R^3\to\R$ is a potential function, $f\in C(\R^3\times\R,\R)$ and $a,b>0$ are constants, has been widely studied by many researchers and arises in an interesting physical meaning. Indeed, if we replace $\R^3$ by a bounded domain $\Omega\subset\R^3$ and set $V(x)=0$ in \eqref{1}, then it becomes the following Kirchhoff Dirichlet problem
\begin{equation}\label{12}
\begin{cases}
-\left(a+b\int_{\Omega}|\nabla u|^2dx\right)\Delta u =f(x,u),&x\in\Omega,\\
u=0,&x\in\partial\Omega,
\end{cases}
\end{equation}
which is related to the stationary analogue of the equation
\begin{equation}\label{133}
u_{tt}-\left(a+b\int_{\Omega}|\nabla u|^2dx\right)\Delta u =f(x,u),x\in\Omega,
\end{equation}
presented by Kirchhoff in \cite{Kir} as an extension of the classical D'Alembert’s wave equations for free vibration of a stretched string or membrane, particularly, taking into account the changes in string length caused by oscillations, where $u$ denotes the displacement, $f(x,u)$ is the external force and $a$ is the initial tension while $b$ is related to the intrinsic properties of the string (such as Young's modulus). Similar nonlocal problems also appear in other fields as biological systems, where $u$ describes a process that depends on the average of itself, for example, population density. For more physical background, we refer the readers to the papers\cite{MB,AAS} and the references therein.  After J.L. Lions \cite{Lions} proposed a functional analysis approach to the equation \eqref{133},  the Kirchhoff
Dirichlet problem \eqref{12} received much attention, see \cite{TXH,kz,zzt,AF} and the references therein.
\par
Mathematically, equation \eqref{1} is often referred to as a nonlocal problem as the appearance of the term $(\int_{\R^3}|\nabla u|^2dx)\Delta u$, which implies that equation \eqref{1} is no longer a pointwise identity. This phenomenon causes some mathematical difficulties, which makes tmake the study of \eqref{1} particularly interesting. The qualitative analysis of nontrivial solutions for equation \eqref{1}, including the existence, multiplicity of positive solutions and of sign-changing solutions, has been extensively studied in the last decade, see \cite{Zou,DYB1,gn,TXH1} and the references therein.
\par
More recently, Sun and Wu \cite{SJT1} considered the following scalar case of \eqref{11}
\begin{equation}\label{14}
\begin{cases}
-\left(a+b\int_{\R^3}|\nabla u|^2dx\right)\Delta u+\lambda V(x)u=f(x,u),\hspace{1ex}x\in\R^N,\\
u\in H^1(\R^N),
\end{cases}
\end{equation}
where $N\geq 3$ and $f(x,s)$ is the potential $V$ satisfies the following conditions:
\vskip4mm
\par\noindent
\begin{tabular}{@{}rp{14.80cm}}
	$(V_1)$& $V\in C(\R^N,[0,\infty))$, $\Omega=\textup{int }V^{-1}(0)$ is nonempty and has smooth boundary with $\overline{\Omega}=V^{-1}(0)$;\\
	$(V_2)$& there exists $c>0$ such that the set $\{x\in\R^N|V(x)<c\}$ is nonempty and has finite Lebesgue measure.\\
\end{tabular}
\vskip4mm
\par
By assuming different conditions on the nonlinear term $f(x,u)$, they obtained the existence and the nonexistence of nontrivial solutions of \eqref{14} by using variational methods. Moreover, the authors also explored the asymptotic behavior of nontrivial solutions with the help of the vanishing lemma \cite{Lions1}.
Very recently, Zhang and Du  \cite{zhang} studied \eqref{14} when $N=3$ and $f(x,u)=|u|^{p-2}u$ with $2<p<4$ and proved the existence and asymptotic behavior of positive solutions by combining the truncation technique and the parameter-dependent compactness condition. For more results about Kirchhoff type problems like \eqref{14}, we refer the readers to \cite{DYB,SJT,SJT2} and the references therein.
\par
We observe that there exist extensive papers  in the study of the coupled elliptic systems on the whole space, we refer the readers to \cite{Lv,FSX,Lv1,SHX,cgf,Wu,zhou} and the references therein. Let us state some
known results. When $a_1=a_2=1$, $b_1=b_2=0$  and the three-dimensional space $\R^3$ is replaced by $\R^N$, the system \eqref{11} is reduced to the following semilinear Schr\"{o}dinger system:
\begin{equation}
\begin{cases}
-\Delta u+\lambda V(x)u=\frac{\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},&x\in\R^N,\\
-\Delta v+\lambda W(x)v=\frac{\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,&x\in\R^N,\\
u,v\in \mathcal{D}^{1,2}(\R^N),
\end{cases}
\end{equation}
where $N\geq 3$, $\lambda>0$ is a parameter, $\alpha, \beta>1$ and $\alpha+\beta<2^*=2N/(N-2)$. $V(x)$ and $W(x)$ satisfy $(H_1)$ and $(H_2)$.
In \cite{FSX}, Furtado, Silva and Xavier studied the existence and multiplicity of solutions  when the parameter $\lambda$ is large enough. Moreover, they also studied the asymptotic behavior of these solutions when $\lambda\to\infty$ via the Nehari manifold on the limit system.
Shi and Chen \cite{SHX} considered system \eqref{11} when $\lambda=1$ and the nonlinear terms are replaced by $\mu F_u(x,u,v)+|u|^{\tau-2}u$ and $\mu F_v(x,u,v)+|v|^4v$, respectively, where $4<\tau<6$ and $\mu>0$ is a parameter. The authors proved  the existence of ground state solutions when $V(x)=W(x)$ is a continuous and asymptotically periodic potential functional and the nonlinear term $F(x,u,v)$ is 4-superlinear at infinity and satisfies the following monotonicity condition, i.e., 
\vskip4mm
\par\noindent
\begin{tabular}{@{}rp{14.80cm}}
	$(F_1)$& $\frac{F(x,u,v)}{|(u,v)|^4}\to+\infty$ uniformly in $x\in\R^3$ as $(u,v)\to+\infty$ ;\\
	$(F_2)$& $u\to \frac{F_u(x,u,v)}{|u|^3}$ and $v\to \frac{F_v(x,u,v)}{|v|^3}$ are nondecreasing on $(-\infty,0)\cup(0,+\infty)$.\\
\end{tabular}
\vskip4mm
\par
L\"{u} and Xiao \cite{Lv} investigated the following coupled Kirchhoff-type systems 
\begin{equation}
\begin{cases}
-\left(a+b\int_{\R^3}|\nabla u|^2dx\right)\Delta u+\lambda V(x)u=\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},&x\in\R^3,\\
-\left(a+b\int_{\R^3}|\nabla v|^2dx\right)\Delta v+\lambda W(x)v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,&x\in\R^3,
\end{cases}
\end{equation}
where $\alpha,\beta>2$ with $\alpha+\beta<2^*=6$ and the potentials $V$ and $W$ satisfy the following conditions:
\vskip4mm
\par\noindent
\begin{tabular}{@{}rp{14.80cm}}
	$(H'_1)$ & $V(x),W(x)\in C(\R^3,[0,\infty))$. $\Omega=\textup{int } V^{-1}(0)=\textup{int } W^{-1}(0)$ is nonempty with smooth boundary and $\overline{\Omega}=V^{-1}(0)=W^{-1}(0)$;\\
	$(H'_2)$&There exist  $M_1,M_2>0$ such that the sets $\{x\in\R^3:V(x)\leq M_1\}$ and $\{x\in\R^3:W(x)\leq M_2\}$ have finite positive Lebesgue measures.   \\
\end{tabular}
\vskip4mm
\par
By using Nehari manifold and Mountain Pass Theorem, they obtain the existence and multiplicity of nontrivial vector solutions when the parameter $\lambda$ is  large enough, but don't study the asymptotic behavior of nontrivial vector solutions. In \cite{Lv}, the assumption $4<\alpha+\beta$ is crucial to prove the boundedness of Palais-Smale sequences, which implies the 4-superlinear of the nonlinear term at infinity.
\par
However, to the best knowledge of us, the existence of positive vector solutions for \eqref{11} were established only in the case where $\alpha,\beta>2$ with $\alpha+\beta<6$, and there are no existence  results of positive vector solutions for \eqref{11} in the case where  $\alpha,\beta>1$ with $\alpha+\beta\le4$. We wish to point out that the one key difficulty in studying \eqref{11} comes from the fact that standard variational methods cannot be applied directly, even restricting the energy functional $\mathcal{J}_{\bf{b},\lambda}$(see \eqref{16} below) on the Nehari manifold, because Palais-Smale sequences cannot be bounded. Moreover, the fact that the nonlinear term $F(x,u,v)=|u|^\alpha|v|^\beta$ of \eqref{11} does not satisfy the condition $(F_2)$ prevents us from using Nehari manifold and fibering methods. There is another difficulty that the work space is a subspace of $\mathcal{D}^{1,2}(\R^3)\times\mathcal{D}^{1,2}(\R^3)$, but not of $H^{1}(\R^3)\times H^{1}(\R^3)$, which prevents us from applying Lions' vanishing lemma to study the asymptotic behavior of the solution, Therefore, the methods of \cite{zhang,SHX,Lv,SJT1} are not applicable for \eqref{11} when $2<\alpha+\beta\le4$.
\par
Motivated by the works described above, the purpose of this paper is to study the strongly coupled Kirchhoff-type system \eqref{11} in the case where $\alpha,\beta>1$ with $\alpha+\beta\le4$. More specifically, we shall study the existence of positive vector solutions for \eqref{11} in this case. Moreover, we also explore the asymptotic behavior of these solutions as $\bf{b}\to0$ and $\lambda\to \infty$.
\par
It is worth mentioning that the conditions $(H'_1)$ and $(H'_2)$ are stronger than $(H_1)$ and $(H_2)$, respectively. They are both two different equivalent forms of the conditions $(V_1)$ and $(V_2)$ if we let $V(x)=W(x)$. In \cite{SJT1}, 
the condition $(V_2)$ is crucial to prove the continuity of the work space embedded into the Sobolev space $H^1(\R^3)$, which can be applied directly in \cite{Lv} under the condition $(H'_2)$, but not in the present paper under the condition $(H_2)$. Hence, we don't expect to find positive vector solutions for \eqref{11} in $H^1(\R^3)$. However, by virtue of the strong coupling of the system and the condition $(H_2)$, we can apply variational methods to study \eqref{11} by considering the energy functional $\mathcal{J}_{\bf{b},\lambda}$ defined in an appropriate closed subspace of $\mathcal{D}^{1,2}(\R^3)\times\mathcal{D}^{1,2}(\R^3)$.
\par
To state our main results, we introduce the following closed subspaces of $\mathcal{D}^{1,2}(\R^3)$ associated with the potentials $V$ and $W$:\[E_{V}=\left\{u\in \mathcal{D}^{1,2}(\R^3):\int_{\R^3}V(x)u^2dx<+\infty \right\}\]
and
\[E_{W}=\left\{u\in \mathcal{D}^{1,2}(\R^3):\int_{\R^3}W(x)u^2dx<+\infty \right\}.\]
Thus the natural space in this paper is the Hilbert space $E=E_{V}\times E_{W}$ with the inner product and norm
\[\langle z,\zeta\rangle=\int_{\R^3}\left[\nabla u\nabla\varphi+\nabla v\nabla\psi+V(x)u\varphi+W(x)v\psi\right]dx,\hspace{2ex} ||z||=\langle z,z\rangle^{1/2}\]
for any $z=(u,v)\in E$ and $\zeta=(\varphi, \psi)\in E$. Throughout this paper, without loss of generality, we always assume that $a_1=a_2=1$ in \eqref{11}. Given $\lambda>0$, we let  $E_\lambda=(E,||\cdot||_{\lambda})$ be equipped with the inner product and norm 
\[\langle z,\zeta\rangle_\lambda=\int_{\R^3}\left[\nabla u\nabla\varphi+\nabla v\nabla\psi+\lambda V(x)u\varphi+\lambda W(x)v\psi\right]dx,\hspace{2ex} ||z||_\lambda=\langle z,z\rangle_\lambda^{1/2}.\]
It is easy to see that $||z||\leq||z||_{\lambda}$ for $\lambda\geq1$.
\par
We shall  prove the existence of positive vector solutions of \eqref{11}, to do this we consider $\mathcal{J}_{{\bf{b}},\lambda}:E_\lambda\to\R$ given by 
\begin{equation}\label{16}
\mathcal{J}_{\bf{b},\lambda}(z)=\frac{1}{2}||z||_\lambda^2+\frac{1}{4}(b_1|\nabla u|^4_2+b_2|\nabla v|^4_2)-\frac{1}{\alpha+\beta}\int_{ \R^3}|u^+|^\alpha|v^+|^\beta dx.
\end{equation}
In view of the conditions $(H_1)$ and $(H_2)$, the functional $\mathcal{J}_{\bf{b},\lambda}$ is well defined and of class $C^1$. Moreover, it is standard to see that \eqref{11} is variational and its solutions are the critical points of \eqref{16}.
\par
In our first result we study the existence of positive solutions of \eqref{11}. Here we observe that we call $z=(u,v)$ a positive vector function if the functions  $u$ and $v$ are positive a.e. in $\R^3$. We also observe that, if $z=(u,v)\in H_0^1(\Omega_1)\times H_0^1(\Omega_2)$ is a nontrivial vector solution of \eqref{19}, then by the condition $(H_1)$, $z$ is also a nontrivial vector solution of \eqref{11} for any $\lambda>0$. Therefore, we are interested in  positive vector solution of \eqref{11}, and obviously it does not lie in $H_0^1(\Omega_1)\times H_0^1(\Omega_2)$. Moreover, it is reasonable to regard $b_1$ and $b_2$ as parameters in \eqref{11}, since $b_1$ and $b_2$ are related to the intrinsic properties of the string, as already mentioned before.
\vskip4mm
\begin{theorem}\label{the1}
	Suppose that  $\alpha$, $\beta>1$ with $\alpha+\beta\le4$ and conditions $(H_1)$--$(H_2)$ hold. Then there exist four numbers $\Lambda^*>1$, $b_*>0$ and $\delta$, $T>0$ such that for any $\lambda>\Lambda^*$ and $0<b_1+b_2<b_*$, the system \eqref{11} possesses at least one positive vector solution $z_{\bf{b},\lambda}=(u_{\bf{b},\lambda},  v_{\bf{b},\lambda})\in E_\lambda$  satisfying
	\begin{equation}\label{13}
	\delta\le||z_{\bf{b},\lambda}||_\lambda\le T,\text{ for any } b_1,b_2 \text{ and } \lambda.
	\end{equation} 
\end{theorem}
\begin{remark}
	\rm{}\eqref{11} has no nontrivial vector solution in $H_0^1(\Omega_1)\times H_0^1(\Omega_2)$ when $b_1+b_2$ is large enough. Indeed, if $z=(u,v)\in H_0^1(\Omega_1)\times H_0^1(\Omega_2)$ is a nontrivial vector solution of \eqref{11}, then similar to the proof of Lemma \ref{L21}, we have 
	\[\begin{aligned}
	0=&|\nabla u|_{2,\Omega_1}^2 + |\nabla v|_{2,\Omega_2}^2+b_1|\nabla u|_{2,\Omega_1}^4+b_2|\nabla v|_{2,\Omega_2}^4-\int_{\Omega_1\cap\Omega_2}|u|^\alpha|v|^\beta dx\\
	\geq&|\nabla u|_{2,\Omega_1}^2 + |\nabla v|_{2,\Omega_2}^2+b_1|\nabla u|_{2,\Omega_1}^4+b_2|\nabla v|_{2,\Omega_2}^4-C\left(|\nabla u|_{2,\Omega_1}^2 + |\nabla v|_{2,\Omega_2}^2\right)^{\frac{\alpha+\beta}{2}},\\
	\end{aligned}\]
	where $C=S^{-\frac{\alpha+\beta}{2}}|\Omega_1\cap\Omega_2|^{\frac{6-\alpha-\beta}{6}}>0$ and $|\cdot|$ is the Lebesgue measure. This is a contradiction for $\max\{b_1,b_2\}$ large enough.
\end{remark}
Now we give our main ideas for the proof of Theorem \ref{the1}. To overcome the obstacle of finding bounded Palais-Smale sequence for $\mathcal{J}_{\bf{b},\lambda}$, as in \cite{SJP}, we use a cut-off function $\xi\in C^\infty(\R_+,[0,1])$ satisfying
\begin{equation}\label{199}
\begin{cases}
\xi(t)=1,&t\in[0,1],\\
\xi(t)=0,&t\in[2,+\infty),\\
\xi'(t)\le0,&t\in(0,\infty),\\
||\xi'||_\infty\le2,
\end{cases}
\end{equation}
and  move to study the following truncated functional $\mathcal{J}_{\bf{b},\lambda}^T$: $E_\lambda\to\R$ defined by
\begin{equation}\label{110}
\mathcal{J}_{\bf{b},\lambda}^T(z)=\frac{1}{2}||z||_\lambda^2+\frac{1}{4}\xi\left(\frac{||z||_\lambda^2}{T^2}\right)(b_1|\nabla u|^4_2+b_2|\nabla v|^4_2)-\frac{1}{\alpha+\beta}\int_{ \R^3}|u^+|^\alpha|v^+|^\beta dx,
\end{equation}
for any $T>0$. Firstly, by the Mountain Pass Theorem without (PS) condition \cite{mw},, we can obtain a  Palais-Smale sequence $\{z_n\}=\{(u_n,v_n)\}$ of $\mathcal{J}_{\bf{b},\lambda}^T$ at the mountain pass level $c_{\bf{b},\lambda}^T$ for small $b_1+b_2>0$. Secondly, we have an important upper bound for $c_{\bf{b},\lambda}^T$ that is independent of $T$, $b_1,b_2$ and $\lambda$. By choosing an appropriate $T>0$, we can deduce that, up to a subsequence, $||z_n||_\lambda\le T$ for all $n\in\mathbb{N}$ by restricting $b_1+b_2>0$ small enough, and so $\{z_n\}$ is a bounded Palais-Smale sequence of $\mathcal{J}_{\bf{b},\lambda}$, i.e.,   
\par\noindent
$$\sup\limits_{n\in\mathbb{N}}||z_n||_{\lambda}\le T,\hspace{1ex}\mathcal{J}_{\bf{b},\lambda}(z_n)\to c^{T}_{\bf{b},\lambda}\hspace{1ex}\text{and}\hspace{1ex}||\mathcal{J}_{\bf{b},\lambda}'(z_n)||_{E^*_\lambda}\to 0,\hspace{1ex}\text{as}\hspace{1ex}n\to\infty,$$
where $E^*_\lambda$ is the dual space of $E_\lambda$. Finally, to recover the compactness, inspired by \cite{BW1}, we establish the parameter-dependent compactness condition to prove $\mathcal{J}_{\bf{b},\lambda}$ satisfies $\textup{(PS)}_{c^{T}_{\bf{b},\lambda}}$ condition for $\lambda>0$ large enough. 
\par
In our next results, we study the asymptotic behavior of the positive vector solutions obtained by Theorem \ref{the1} as $\bf{b}\to\bf{0}$ and $\lambda\to\infty$. It is worth emphasizing there are two usual methods to explore the asymptotic behavior of solutions, one is with the help of the Lions' vanishing lemma \cite{Lions,mw}, see e.g. \cite{BPW,SJT,zhang,zhao}, the other is via the Nehari manifold on the limit system, see \cite{FSX}. As already mentioned before, the Nehari method is not applicable in our case. Moreover, the work space $E_\lambda$ is a subspace of $\mathcal{D}^{1,2}(\R^3)\times\mathcal{D}^{1,2}(\R^3)$, but not of $H^{1}(\R^3)\times H^{1}(\R^3)$, which prevents us from applying Lions' vanishing lemma. By Lemma \ref{L21}, the key inequalities \eqref{46}  and \eqref{38} are obtained, allowing us to overcome this difficulty.
\begin{theorem}\label{the2}
	Let $z_{\bf{b},\lambda}$ be the positive vector solutions of \eqref{11} obtained by Theorem \ref{the1}. Then for any $b_1+b_2\in (0,b_*)$ fixed, $z_{\bf{b},\lambda}\to z_{\bf{b}}$ in $E$ as $\lambda\to\infty$, where $z_{\bf{b}}=(u_{\bf{b}},v_{\bf{b}})\in H_0^1(\Omega_1)\times H_0^1(\Omega_2)$ is a positive vector solution of
	\begin{equation}\label{19}
	\begin{cases}
	-\left(1+b_1\int_{\Omega_1}|\nabla u|^2dx\right)\Delta u=\frac{\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},&x\in\Omega_1,\\
	-\left(1+b_2\int_{\Omega_2}|\nabla v|^2dx\right)\Delta v=\frac{\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,&x\in\Omega_2,\\
	u\in H_0^1(\Omega_1),v\in H^1_0(\Omega_2).
	\end{cases}\tag{$\mathcal{K}_{\bf{b},\infty}$}
	\end{equation}
\end{theorem}
\begin{theorem}\label{the3}
	Let $z_{\bf{b},\lambda}$ be the positive vector   solutions of \eqref{11} obtained by Theorem \ref{the1}. Then for any $\lambda\in (\Lambda^*, \infty )$ fixed, $z_{\bf{b},\lambda}\to z_\lambda$ in $E_\lambda$ as $\bf{b}\to\bf{0}$, where $z_\lambda\in E_\lambda$ is a positive vector solution of
	\begin{equation}\label{3res}
	\begin{cases}
	-\Delta u+\lambda V(x)u=\frac{\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},&x\in\R^3,\\
	-\Delta v+\lambda W(x)v=\frac{\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,&x\in\R^3,\\
	u,v\in \mathcal{D}^{1,2}(\R^3).
	\end{cases}\tag{$\mathcal{K}_{\bf{0},\lambda}$}
	\end{equation} 
\end{theorem}
\begin{theorem}\label{the4}
	Let $z_{\bf{b},\lambda}$ be the positive vector   solutions of \eqref{11} obtained by Theorem \ref{the1}. Then $z_{\bf{b},\lambda}\to z_{\bf{0}}$ in $E$ as  $\bf{b}\to\bf{0}$ and $\lambda\to\infty$, where $z_{\bf{0}}=(u_{\bf{0}},v_{\bf{0}})\in H_0^1(\Omega_1)\times H_0^1(\Omega_2)$ is a positive vector solution of 
	\begin{equation}\label{4res}
	\begin{cases}
	-\Delta u=\frac{\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},&x\in\Omega_1,\\
	-\Delta v=\frac{\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,&x\in\Omega_2,\\
	u\in H_0^1(\Omega_1),v\in H^1_0(\Omega_2).
	\end{cases}\tag{$\mathcal{K}_{0,\infty}$}
	\end{equation}
\end{theorem}
\begin{remark}
	\text{}\\
	\begin{tabular}{@{}rp{15.05cm}}
		\rm{(\romannumeral1)}&\rm{}Let $b_1(\text{or } b_2)>0$ be a small fixed parameter, similar to the proof of Theorem \ref{the2}, we can also obtain the asymptotic behavior of these solutions obtained by Theorem \ref{the1} and  as $b_2(\text{or } b_1)\to0$ and  $\lambda\to\infty$.\\
		\rm{(\romannumeral2)}&\rm{}Let $b_1(\text{or } b_2)>0$ be a small fixed parameter and $\lambda>0$ be a large fixed-paramete, similar to the proof of Theorem \ref{the3}, we can also obtain the asymptotic behavior of these solutions obtained by Theorem \ref{the1} as $b_2(\text{or } b_1)\to0$.\\
	\end{tabular}
\end{remark}
\par
The remainder of this paper is organized as follows. In Section \ref{sec2}, we give some preliminary results which are crucial throughout the paper. we prove Theorem \ref{the1} in Section \ref{sec3}. The proofs of Theorems \ref{the2}, \ref{the3} and \ref{the4} will be given in Section \ref{sec4}.
\par
Throughout the paper, we use the following notations:\par
$\bullet$ $H^1(\R^3)$ is the usual Sobolev space equipped with the inner product abd norm
$$(u,v)=\int_{\R^3}(\nabla u\nabla v)+uv)dx;\hspace{2ex}||u||^2=\int_{ \R^3}(|\nabla u|^2+u^2)dx.$$\par
$\bullet$ $\mathcal{D}^{1,2}(\R^3) $ is the completion of $C^{\infty}_0(\R^3)$ with respect to the norm 
$$||u||_{\mathcal{D}^{1,2}}^2=|\nabla u|_2^2=\int_{ \R^3}|\nabla u|^2dx.$$\par
$\bullet$ $L^r(\Omega)$, $1\le r\le\infty$, $\Omega\subset\R^3$, denotes a Lebesgue space,  the norm in $L^r(\R^3)$ is denoted by\par \hspace{1.25ex} $|u|_{r,\Omega}$,  where $\Omega$ is a proper subset of $\R^3$, by $|u|_r$ where $\Omega=\R^3$.\par
$\bullet$ $S$ is the best constant for the embedding of  $\mathcal{D}^{1,2}(\R^3)\hookrightarrow$ $ L^6(\R^3)$, i.e., 
\begin{equation}\label{S}
|u|_6\le S^{-\frac{1}{2}}||u||_{\mathcal{D}^{1,2}}\quad \forall u\in \mathcal{D}^{1,2}(\R^3).
\end{equation}\par
$\bullet$ For a mensurable function $u$, we denote by $u^+$ and $u^-$ its positive and negative parts\par
\hspace{1.25ex} respectively, given by
\begin{equation*}
u^+=\max\{u,0\},\hspace{1ex} u^-=\max\{-u,0\}.\end{equation*}\par 
$\bullet$ $\rightarrow$ and $\rightharpoonup$ denote the strong and weak convergence in the related function space respectively.\par
$\bullet$ $o(1)$ denotes any quantity which tends to zero when $n\to\infty$.\par
$\bullet$ Let $E$ be a real  Banach space and $I\in C^1(E,\R)$. We call that $I$ satisfies the Palais-Smale\par \hspace{1.25ex} condition at $c$ ($\textup{(PS)}_{c}$ condition for short), if any sequence  $\{u_n\}\subset E$ satisfying  $I(u_n)\to c$\par \hspace{1.25ex}  and $||I'(u_n)||_{E^*}\to0$ possesses a convergent subsequence.
\vskip4mm
{\section{Preliminary results}\label{sec2}}
\setcounter{equation}{0}
\vskip2mm
In this section, we give some preliminary lemmas for the proof of Theorem \ref{the1}. It is notable that the following lemma allows us to show the (PS) condition of $\mathcal{J}_{{\bf{b}},\lambda}$ in another way.
\begin{lemma}\label{L21}
	Suppose that  $\alpha$, $\beta>1$ with $\alpha+\beta\le4$ and $(H_1)$--$(H_2)$ hold. Then for any $z=(u,v)\in E_{\lambda}$ and $\lambda>0$, there exists a constant $\hat{c}>0$, independent of $\lambda$, such that 
	\begin{equation*}
	\int_{ \R^3}|u|^\alpha|v|^\beta dx \leq \hat{c} ||z||^{\alpha+\beta}.
	\end{equation*}
\end{lemma}
\begin{proof}
	We borrow an idea from \cite[Lemmas 2.1-2.2]{FSX}. It follows from  $(H_2)$ that
	\begin{equation}\label{21}
	\begin{aligned}
	\int_{\R^3}|uv|dx&\le \int_{\mathcal{M}}|uv|dx+\frac{1}{ c}\int_{{\mathcal{M}}^{c}}( V(x)u^2)^{\frac{1}{2}}( W(x)v^2)^{\frac{1}{2}}dx\\
	&\le |u|_6|v|_6|\mathcal{M}|^{\frac{2}{3}}+\frac{1}{c}
	\left(\int_{{\mathcal{M}}^{c}} V(x)u^2dx\right)^{\frac{1}{2}}\left(\int_{{\mathcal{M}}^{c}}W(x)v^2dx\right)^{\frac{1}{2}}\\
	&\le S^{-1}|\nabla u|_2|\nabla v|_2|\mathcal{M}|^{\frac{2}{3}}+\frac{1}{2 c}\int_{\R^3}V(x)u^2+ W(x)v^2dx\\
	&\le \frac{1}{2}\max\left\{S^{-1}|\mathcal{M}|^{\frac{2}{3}}, \frac{1}{ c} \right\}||z||^2,
	\end{aligned}
	\end{equation}
	where we have used the H\"{o}lder inequality and \eqref{S}. 
	Taking $r=\frac{6}{8-\alpha-\beta}$, then we can obtain that
	\begin{equation}\label{22}
	\frac{\alpha-1}{6}+\frac{\beta-1}{6}+\frac{1}{r}=1.
	\end{equation}
	Since $1<r\le\frac{3}{2}$, there exists $\theta=\frac{\alpha+\beta-2}{16-2(\alpha+\beta)}\in (0,\frac{1}{4}]$ such that $r=2\theta+1$. We can deduce from H\"{o}lder inequality and \eqref{S} that
	\begin{equation}\label{23}
	\begin{aligned}
	\int_{\R^3}|uv|^rdx&\le \left(\int_{\R^3}|uv|^3dx\right)^{\theta}\left(\int_{\R^3}|uv|dx\right)^{1-\theta}\\
	&\le \left[\frac{1}{2}\left(|u|_6^6+|v|^6_6\right) \right]^\theta\left(\int_{\R^3}|uv|dx\right)^{1-\theta}\\
	&\le \left[\frac{1}{2}S^{-3}\left(|\nabla u|_2^6+|\nabla v|^6_2\right) \right]^\theta\left(\int_{\R^3}|uv|dx\right)^{1-\theta}\\
	&\le S^{-3\theta}||z||^{6\theta}\left(\int_{\R^3}|uv|dx\right)^{1-\theta}.
	\end{aligned}
	\end{equation}
	From \eqref{21}--\eqref{23}, we can derive 
	\begin{equation}\label{24}
	\begin{aligned}
	\int_{\R^3}|u|^\alpha |v|^\beta dx&=\int_{\R^3}|u|^{\alpha-1} |v|^{\beta-1}|uv| dx\\
	&\le
	\left(\int_{\R^3}|u|^6dx\right)^\frac{\alpha-1}{6}\left(\int_{\R^3}|v|^6dx\right)^\frac{\beta-1}{6}\left(\int_{\R^3}|uv|^r\right)^\frac{1}{r}\\
	&\le S^{-\frac{\alpha+\beta-2}{2}}||z||^{\alpha+\beta-2}\left(\int_{\R^3}|uv|^r\right)^\frac{1}{r}\\
	&\le S^{-\frac{9\theta}{2\theta+1}}||z||^{\alpha+\beta-2+\frac{6\theta}{r}}\left(\int_{\R^3}|uv|dx\right)^{\frac{1-\theta}{r}}\\
	&\le \left(\frac{1}{2}\max\left\{S^{-1}|\mathcal{M}|^{\frac{2}{3}}, \frac{1}{c} \right\}\right)^{\frac{1-\theta}{2\theta+1}}S^{-\frac{9\theta}{2\theta+1}}||z||^{\alpha+\beta}\\
	&= \hat{c}||z||^{\alpha+\beta},
	\end{aligned}
	\end{equation}
	where $\hat{c}=\left(\frac{1}{2}\max\left\{S^{-1}|\mathcal{M}|^{\frac{2}{3}}, \frac{1}{c} \right\}\right)^{\frac{1-\theta}{2\theta+1}}S^{-\frac{9\theta}{2\theta+1}}$ is independent of $\lambda$. The proof is complete.
\end{proof}
The following lemma shows that the the coupled term $\int_{ \R^3}|u|^\alpha|v|^\beta dx$ has BL-splitting property, which is an another version of Br\'{e}zis-Lieb lemma \cite{BL}.
\begin{lemma}(\label{BL}\cite[Lemma 4.2]{FSX})
	Let $\{z_n\}=\{(u_n,v_n)\}\subset E_\lambda$ be such that $(u_n, v_n)\rightharpoonup(u,v)$ in $E_\lambda$. Then
	$$\lim\limits_{n\to\infty}\int_{ \R^3}(|u_n|^\alpha|v_n|^\beta-|u_n-u|^\alpha|v_n-v|^\beta)dx=\int_{ \R^3}|u|^\alpha|v|^\beta dx.$$
\end{lemma}
\par
To overcome the obstacle of finding bounded Palais-Smale sequence for $\mathcal{J}_{\bf{b},\lambda}$, we first define a cut-off function $\xi\in C^\infty(\R_+,[0,1])$ (see \eqref{199} above). For any $T>0$, we consider the truncated functional $\mathcal{J}_{\bf{b},\lambda}^T:E_\lambda\to\R$ 
\begin{equation}
\mathcal{J}_{\bf{b},\lambda}^T(z)=\frac{1}{2}||z||_\lambda^2+\frac{1}{4}\xi\left(\frac{||z||_\lambda^2}{T^2}\right)\left(b_1|\nabla u|_2^4+b_2|\nabla v|_2^4\right)-\frac{1}{\alpha+\beta}\int_{ \R^3}|u^+|^\alpha|v^+|^\beta dx.
\end{equation}
It is easy to see that $\mathcal{J}_{\bf{b},\lambda}^T$ is of class $C^1$ with derivative
\begin{equation}\label{26}
\begin{aligned}
\langle (\mathcal{J}_{\bf{b},\lambda}^T)'(z),\zeta\rangle=&
\langle z,\zeta\rangle_\lambda+\frac{1}{2T^2}\xi'\left(\frac{||z||_\lambda^2}{T^2}\right)\left(b_1|\nabla u|_2^4+b_2|\nabla v|_2^4\right)\langle z,\zeta\rangle_\lambda\\
&+\xi\left(\frac{||z||_\lambda^2}{T^2}\right)\left(b_1|\nabla u|_2^2\int_{ \R^3}\nabla u\nabla \varphi dx +b_2|\nabla v|_2^2\int_{ \R^3}\nabla v\nabla\psi dx \right)\\
&-\frac{\alpha}{\alpha+\beta}\int_{\R^3}|u^+|^{\alpha-2}u^+\varphi|v^+|^\beta dx-\frac{\beta}{\alpha+\beta}\int_{\R^3}|u^+|^{\alpha}|v^+|^{\beta-2}v^+\psi dx,
\end{aligned}
\end{equation}
for any $z=(u,v)\in E_\lambda$ and $\zeta=(\varphi, \psi)\in E_\lambda$.
With this penalization, for a properly chosen $T>0$ and $b_1+b_2$ small enough, we are able to find a Palais-Smale sequence $\{z_n\}$ of $\mathcal{J}_{\bf{b},\lambda}^T$ satisfying $||z_n||_\lambda\le T$ and so $\{z_n\}$ is also a Palais-Smale sequence of $\mathcal{J}_{\bf{b},\lambda}$ satisfying $||z_n||_\lambda\le T$.
\par
The following Lemmas \ref{L233}--\ref{L244} imply that $\mathcal{J}_{\bf{b},\lambda}^T$ has the mountain pass geometry.
\begin{lemma}\label{L233}
	Suppose that $\alpha>1$, $\beta>1$ with $\alpha+\beta\le 4$ and $(H_1)$--$(H_2)$ hold. Then there exist $\rho, \eta>0$ such that for any $b_1$, $b_2$, $T>0$ and $\lambda\geq1$, there holds
	\[\inf\{\mathcal{J}^T_{\bf{b},\lambda}:z\in E_\lambda\hspace{1ex} \textup{with}\hspace{1ex} ||z||_{\lambda}=\rho\}\geq\eta.\]
\end{lemma}	
\begin{proof}
	It follows from Lemma \ref{L21} that
	\[\mathcal{J}_{\bf{b},\lambda}^T(z)\geq\frac{1}{2}||z||_\lambda^2-\frac{\hat{c}}{\alpha+\beta}||z||_\lambda^{\alpha+\beta}=||z||_\lambda^2\left(\frac{1}{2}-\frac{\hat{c}}{\alpha+\beta}||z||_\lambda^{\alpha+\beta-2}\right),\]
	where $\hat{c}$ is given in Lemma \ref{L21} and independent of $b_1$, $b_2$, $T$ and $\lambda.$ Consequently, by choosing $\rho=\left(\frac{\alpha+\beta}{4\hat{c}}\right)^{\frac{1}{\alpha+\beta-2}}>0$ and $\eta=\frac{1}{4}\rho^2$, we arrive at the desired result.
\end{proof}
\begin{lemma}\label{L244}
	Suppose that $\alpha$, $\beta>1$ with $\alpha+\beta\le4$ and $(H_1)$--$(H_2)$ hold. Then there exist $b^*>0$ and $e=(e_0,e_0)\in C_0^\infty(\Omega_1)\times C_0^\infty(\Omega_2)$ such that for any $T$, $\lambda>0$ and $b_1+b_2\in (0,b^*)$, we have $\mathcal{J}_{\bf{b},\lambda}^T(e)<0$ with $||e||_\lambda>\rho$.
\end{lemma}
\begin{proof}
	We define the functional $\mathcal{I}_\lambda:E_\lambda\to\R$ by
	\[\mathcal{I}_\lambda(z)=\frac{1}{2}||z||_\lambda^2-\frac{1}{\alpha+\beta}\int_{ \R^3}|u^+|^\alpha|v^+|^\beta dx.\]
	Let $e_1=e_2\in C_0^\infty(\Omega_1\cap\Omega_2)$, then we have $V(x)e_1=W(x)e_1=0$ on $\R^3$ and $||(e_1,e_1)||_\lambda=\sqrt{2}|\nabla e_1|_2$. Hence, by $\alpha+\beta>2$, we can see that
	\[\lim\limits_{t\to\infty}\mathcal{I}_{\lambda}(t(e_1,e_1))=\lim\limits_{t\to\infty}\left(t^2\int\limits_{\Omega_1\cap\Omega_2}|\nabla e_1|^2dx-\frac{t^{\alpha+\beta}}{\alpha+\beta}\int\limits_{\Omega_1\cap\Omega_2}|e_1|^{\alpha+\beta} dx\right)=-\infty.\]
	So we further have that there exists $e_0\in C_0^\infty(\Omega_1\cap\Omega_2)$ with $|\nabla e_0|_2>\frac{\rho}{\sqrt{2}}$ and $e_0^+\not\equiv0$, independent of $b_1$, $b_2$, $T$ and $\lambda$, such that $\mathcal{I}_{\lambda}(e)\le -1$. Since
	\[\mathcal{J}_{\bf{b},\lambda}^T(e)=\mathcal{I}_{\lambda}(e)+\frac{b_1+b_2}{4}\xi\left(\frac{||e||_\lambda^2}{T^2}\right)|\nabla e_0|_2^4\le -1+\frac{b_1+b_2}{4}|\nabla e_0|_2^4,\]
	there exists  $b^*>0$, independent of $\lambda$ and $T$, such that $\mathcal{J}_{\bf{b},\lambda}^T(e)<0$ for any $\lambda, T>0$ and $b_1+b_2\in(0,b^*)$. The proof is complete.
\end{proof}
By Lemmas \ref{L233}--\ref{L244} and the Mountain Pass Theorem without (PS) condition \cite{mw}, we can obtain that for any $\lambda\geq 1$, $T>0$ and $b_1+b_2\in(0,b^*)$, there is a $\textup{(PS)}_{c^{T}_{\bf{b},\lambda}}$ sequence $\{z_n\}=\{(u_n,v_n)\}\subset E_\lambda$ such that
\begin{equation}\label{27}
\mathcal{J}_{\bf{b},\lambda}^T\to c^{T}_{\bf{b},\lambda}\geq\eta>0\hspace{1ex}\text{and}\hspace{1ex}||(\mathcal{J}_{\bf{b},\lambda}^T)'(z_n)||_{E_\lambda^*}\to 0,\hspace{1ex}\text{as}\hspace{1ex}n\to\infty,
\end{equation}
where 
\[c^{T}_{\bf{b},\lambda}:=\inf\limits_{\gamma \in \Gamma }\max\limits_{t\in[0,1]} \mathcal{J}_{\bf{b},\lambda}^T(\gamma(t)),\]
and  
\[\Gamma=\{\gamma\in C([0,1],E_\lambda);\gamma(0)=0,\gamma(1)=e\}.\]
To find a bounded Palais-Smale sequence,  we have an important upper bound for $c^{T}_{\bf{b},\lambda}$, which is the keystone of the truncation technique.
\begin{lemma}\label{L255}
	Suppose that $\alpha$, $\beta>1$ with $\alpha+\beta\le4$ and the conditions $(H_1)$--$(H_2)$ hold. Then for for any $\lambda\geq 1$, $T>0$ and $b_1+b_2\in(0,b^*)$, there exists $D>0$, independent of $b_1$, $b_2$, $T$ and $\lambda$, such tha $c^{T}_{\bf{b},\lambda}\le D$.
\end{lemma}
\begin{proof}
	It follows from $e_0\in C_0^\infty(\Omega_1\cap\Omega_2)$ that
	\[\mathcal{J}_{\bf{b},\lambda}^T(t(e_0,e_0))\le t^2\int\limits_{\Omega_1\cap\Omega_2}|\nabla e_0|^2dx+\frac{b^*t^4}{4}\left(\int\limits_{\Omega_1\cap\Omega_2}|\nabla e_0|^2dx\right)^2-\frac{t^{\alpha+\beta}}{\alpha+\beta}\int\limits_{\Omega_1\cap\Omega_2}|e_0^+|^{\alpha+\beta} dx.\]
	Hence, by the definition of $c^{T}_{\bf{b},\lambda}$, there exists a constant $D>0$ such that
	\[c^{T}_{\bf{b},\lambda}\le \max\limits_{t\in[0,1]}\mathcal{J}_{\bf{b},\lambda}^T(te)\le D.\]
	The proof is complete.
\end{proof}
\par
The following key lemma shows that for a properly  chosen $T> 0$, up to a subsequence, $\{z_n\}$ is $\textup{(PS)}_{c^{T}_{\bf{b},\lambda}}$ sequence of $\mathcal{J}_{\bf{b},\lambda}^T$ satisfying $||z_n||_\lambda\le T$, and so $\{z_n\}$ is also a bounded $\textup{(PS)}_{c^{T}_{\bf{b},\lambda}}$ sequence of  $\mathcal{J}_{\bf{b},\lambda}$, i.e., 
\begin{equation}\label{277}
\mathcal{J}_{\bf{b},\lambda}\to c^{T}_{\bf{b},\lambda}\geq\eta>0\hspace{1ex}\text{and}\hspace{1ex}||\mathcal{J}_{\bf{b},\lambda}'(z_n)||_{E_\lambda^*}\to 0,\hspace{1ex}\text{as}\hspace{1ex}n\to\infty.
\end{equation}
\begin{lemma}\label{L26}
	Suppose that $\alpha$, $\beta>1$ with $\alpha+\beta\le4$, the conditions $(H_1)$--$(H_2)$ hold, $\{z_n\}=\{(u_n,v_n)\}\subset E_\lambda$ is a $\textup{(PS)}_{c^{T}_{\bf{b},\lambda}}$ sequence of  $\mathcal{J}^T_{\bf{b},\lambda}$, and let $T=\sqrt{\frac{2(\alpha+\beta)(D+1)}{\alpha+\beta-2}}$. Then there exists $b_*\in(0,b^*)$ such that for any $\lambda\geq 1$ and $b_1+b_2\in(0,b_*)$, up to a subsequence, $||z_n||_\lambda\le T$.
\end{lemma}
\begin{proof}
	Arguing indirectly, suppose that $||z_n||_{\lambda}>T$ up to a subsequence. We next divide the proof into two separate cases.
	\par
	Case 1: $T<||z_n||_{\lambda}\le\sqrt{2}T$. In this case, by \eqref{26}--\eqref{27} and $\xi'(t)\le0$ for all $t>0$, we have
	\begin{equation}\label{29}
	\begin{aligned}
	c^{T}_{\bf{b},\lambda}=&\lim\limits_{n\to\infty}\left(\mathcal{J}_{\bf{b},\lambda}^T(z_n)-\frac{1}{\alpha+\beta}\langle (\mathcal{J}_{\bf{b},\lambda}^T)'(z_n),z_n\rangle\right)\\
	=&\lim\limits_{n\to\infty}\left((\frac{1}{2}-\frac{1}{\alpha+\beta})||z_n||^2_{\lambda}-(\frac{1}{\alpha+\beta}-\frac{1}{4})\xi\left(\frac{||z||_\lambda^2}{T^2}\right)\left(b_1|\nabla u_n|_2^4+b_2|\nabla v_n|_2^4\right)\right.\\
	&\phantom{=\;\;}
	\left.-\frac{1}{2(\alpha+\beta)T^2}\xi'\left(\frac{||z||_\lambda^2}{T^2}\right)\left(b_1|\nabla u_n|_2^4+b_2|\nabla v_n|_2^4\right)||z_n||^2_{\lambda}\right)\\
	\geq& \liminf\limits_{n\to\infty}\left((\frac{1}{2}-\frac{1}{\alpha+\beta})||z_n||^2_{\lambda}-(\frac{1}{\alpha+\beta}-\frac{1}{4})\left(b_1+b_2\right)||z_n||^4_{\lambda}\right)\\
	\geq& (D+1)-\frac{4(4-\alpha-\beta)(b_1+b_2)}{(\alpha+\beta-2)^2}(D+1)^2,
	\end{aligned}
	\end{equation}
	which contradicts the fact that  $c^{T}_{\bf{b},\lambda}\le D$ by choosing $b_*$ small enough.
	\par
	Case 2: $||z_n||_{\lambda}>\sqrt{2}T$. In this case, by the similar computation to \eqref{29} and using the fact that
	$\xi\left(\frac{||z||_\lambda^2}{T^2}\right)=\xi'\left(\frac{||z||_\lambda^2}{T^2}\right)=0$, we get
	\begin{equation}
	\begin{aligned}
	c^{T}_{\bf{b},\lambda}=&\lim\limits_{n\to\infty}\left(\mathcal{J}_{\bf{b},\lambda}^T(z_n)-\frac{1}{\alpha+\beta}\langle (\mathcal{J}_{\bf{b},\lambda}^T)'(z_n),z_n\rangle\right)\\
	=&\lim\limits_{n\to\infty}\left((\frac{1}{2}-\frac{1}{\alpha+\beta})||z_n||^2_{\lambda}-(\frac{1}{\alpha+\beta}-\frac{1}{4})\xi\left(\frac{||z||_\lambda^2}{T^2}\right)\left(b_1|\nabla u_n|_2^4+b_2|\nabla v_n|_2^4\right)\notag\right.\\
	&\phantom{=\;\;}
	\left.-\frac{1}{2(\alpha+\beta)T^2}\xi'\left(\frac{||z||_\lambda^2}{T^2}\right)\left(b_1|\nabla u_n|_2^4+b_2|\nabla v_n|_2^4\right)||z_n||^2_{\lambda}\right)\\
	\geq& 2(D+1), 
	\end{aligned}
	\end{equation}
	which is again a contradiction. Hence $||z_n||_{\lambda}\le T$ and the proof is complete.
\end{proof}
\vskip2mm
{\section{Existence of positive vector solutions }\label{sec3}}
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\vskip2mm
To find the critical points of $\mathcal{J}_{\bf{b},\lambda}$, we first  establish the following 
parameter-dependent compactness condition for the functional $\mathcal{J}_{\bf{b},\lambda}$.
\vskip2mm
\begin{proposition}\label{L28}
	Suppose that $\alpha>1$, $\beta>1$ with $\alpha+\beta\le4$ and the conditions $(V_1)$--$(V_2)$ hold, and let $T=\sqrt{\frac{2(\alpha+\beta)(D+1)}{\alpha+\beta-2}}$. Then there exists $\Lambda^*> 1$ such that  $\mathcal{J}_{\bf{b},\lambda}$ satisfies $\textup{(PS)}_{c^{T}_{\bf{b},\lambda}}$ condition in $E_\lambda$ for any $b_1+b_2\in(0,b_*)$ and $\lambda\in(\Lambda^*,\infty)$.
\end{proposition}
\begin{proof}
	Let $\{z_n\}=\{(u_n,v_n)\}\subset E$ be a $\textup{(PS)}_{c^{T}_{\bf{b},\lambda}}$ sequence for $\mathcal{J}_{\bf{b},\lambda}$. From Lemma \ref{L26}, we know that $\{z_n\}$ is bounded in $E_\lambda$, i.e.,
	\[||z_n||_\lambda\le T=\sqrt{\frac{2(\alpha+\beta)(D+1)}{\alpha+\beta-2}}.\]
	Then there exist a subsequence $\{z_n\}$ and $z=(u,v)$ in $E_\lambda$ such that
	\begin{equation}\label{28}
	\begin{cases}
	(u_n,v_n)\rightharpoonup(u,v),& \hspace{1ex}\textup{in} \hspace{1ex}E_\lambda;\\
	(u_n,v_n)\to(u,v),& \hspace{1ex}\textup{a.e. in} \hspace{1ex}\R^3;\\
	(u_n,v_n)\to(u,v),& \hspace{1ex}\textup{in} \hspace{1ex}L_{loc}^{s_1}(\R^3)\times L_{loc}^{s_2}(\R^3),\hspace{1ex} 2\le s_1, s_2<6.\\
	\end{cases}
	\end{equation}
	Now we prove that $z_n\to z$ in $E_\lambda$. Let $\tilde{z}_n=z_n-z$ with $\tilde{u}_n=u_n-u$ and $\tilde{v}_n=v_n-v$. By \eqref{28}, we have 
	\[||z||_\lambda\le \liminf\limits_{n\to\infty}||z_n||_\lambda\le T,\]
	and then
	\begin{equation}\label{211}
	||\tilde{z}_n||_\lambda=||z_n-z||_\lambda\le 2T.
	\end{equation}
	It follows from $(H_2)$ that
	\[\begin{aligned}
	\int_{\R^3}|\tilde{u}_n\tilde{v}_n|dx&=\int_{\mathcal{M}^c}|\tilde{u}_n\tilde{v}_n|dx+\int_{\mathcal{M}}|\tilde{u}_n\tilde{v}_n|dx\\
	&\le\frac{1}{2\lambda c}\int_{\mathcal{M}^c}\lambda V(x)\tilde{u}_n^2+\lambda W(x)\tilde{v}_n^2dx+ |\tilde{u}_n|_{5,\mathcal{M}}|\tilde{v}_n|_{5,\mathcal{M}}|\mathcal{M}|^{\frac{3}{5}}\\
	&\le \frac{1}{2\lambda c}||\tilde{z}_n||_\lambda^2+o(1),
	\end{aligned}\]
	this together with Lemma \ref{L21} implies that
	\begin{equation}\label{2133}
	\begin{aligned}
	\int_{\R^3}|\tilde{u}_n|^\alpha|\tilde{v}_n|^\beta dx
	&\le S^{-\frac{9\theta}{2\theta+1}}||\tilde{z}_n||_\lambda^{\alpha+\beta-2+\frac{6\theta}{2\theta+1}}\left(\int_{\R^3}|\tilde{u}_n\tilde{v}_n|dx\right)^{\frac{1-\theta}{2\theta+1}}\\
	&\le \left(\frac{1}{2\lambda c}\right)^{\frac{1-\theta}{2\theta+1}}S^{-\frac{9\theta}{2\theta+1}}||\tilde{z}_n||_\lambda^{\alpha+\beta}+o(1),
	\end{aligned}
	\end{equation}
	where $\theta=\frac{\alpha+\beta-2}{16-2(\alpha+\beta)}\in (0,\frac{1}{4}]$. 
	Moreover, there exist $A$, $B\in\R$ such that
	\[\int_{ \R^3}|\nabla u_n|^2dx\to A^2,\hspace{2ex}\int_{ \R^3}|\nabla v_n|^2dx\to B^2\]
	and 
	\[\int_{ \R^3}|\nabla u|^2dx\le A^2,\hspace{2ex}\int_{ \R^3}|\nabla v|^2dx\le B^2.\]
	As in \cite{FSX}, it follows from \eqref{28} and the Lebesgue Dominated Convergence Theorem that 
	\begin{equation}\label{212}
	\lim\limits_{n\to\infty}\int_{ \R^3}|u_n|^{\alpha-2}u_n\varphi|v_n|^\beta dx=\int_{ \R^3}|u|^{\alpha-2}u\varphi|v|^\beta dx,\textup{ for any } \varphi\in C_0^\infty(\R^3)
	\end{equation}
	and
	\begin{equation}\label{213}
	\lim\limits_{n\to\infty}\int_{ \R^3}|u_n|^{\alpha}|v_n|^{\beta-2}v_n\psi dx=\int_{ \R^3}|u|^{\alpha}|v|^{\beta-2}v\psi dx,\textup{ for any } \psi\in C_0^\infty(\R^3).
	\end{equation}
	Then by $ \mathcal{J}_{\bf{b},\lambda}'(z_n)\to0$ and \eqref{212}--\eqref{213} , we can get that for any  $\zeta=(\varphi,\psi)\in C_0^\infty(\R^3)\times C_0^\infty(\R^3)$,
	\begin{equation}\label{214}
	\begin{aligned}
	&\langle z,\zeta\rangle_\lambda-\frac{1}{\alpha+\beta}\left(\alpha\int_{\R^3}|u^+|^{\alpha-2}u^+\varphi|v^+|^\beta dx+{\beta}\int_{\R^3}|u^+|^{\alpha}|v^+|^{\beta-2}v^+\psi dx\right) \\
	&+b_1A^2\int_{ \R^3}\nabla u\nabla \varphi dx+b_2B^2\int_{ \R^3}\nabla v\nabla\psi dx=0.
	\end{aligned}
	\end{equation}
	Taking $\zeta=z$ in \eqref{214}, we have 
	\begin{equation}\label{215}
	\begin{aligned}
	||z||^2_\lambda
	+b_1A^2|\nabla u|_2^2+b_2B^2|\nabla v|_2^2-\int_{\R^3}|u^+|^{\alpha}|v^+|^\beta dx =0.
	\end{aligned}
	\end{equation}
	Note that
	\begin{equation}
	\begin{aligned}
	||z_n||^2_\lambda
	+b_1|\nabla u_n|_2^4+b_2|\nabla v_n|_2^4 dx-\int_{\R^3}|u_n^+|^{\alpha}|v_n^+|^\beta dx =o(1).
	\end{aligned}
	\end{equation}
	By the Br\'{e}zis--Lieb Lemma, we have that
	\begin{equation}
	||\tilde{z}_n||_\lambda^2=||z_n||_\lambda^2-||z||_\lambda^2+o(1),
	\end{equation}
	\begin{equation}\label{218}
	|\nabla \tilde{u}_n|_2^2=|\nabla u_n|_2^2-|\nabla u|_2^2+o(1),\hspace{2ex} |\nabla \tilde{v}_n|_2^2=|\nabla v_n|_2^2-|\nabla v|_2^2+o(1).
	\end{equation}
	It follows from \eqref{215}--\eqref{218} and Lemma \ref{BL} that 
	\begin{equation}\label{221}
	\begin{aligned}
	o(1)=&||z_n||^2_\lambda
	+b_1|\nabla u_n|_2^4+b_2|\nabla v_n|_2^4 dx-\int_{\R^3}|u_n^+|^{\alpha}|v_n^+|^\beta dx \\
	&- ||z||^2_\lambda
	-b_1A^2|\nabla u|_2^2-b_2B^2|\nabla v|_2^2+\int_{\R^3}|u^+|^{\alpha}|v^+|^\beta dx \\
	=&||\tilde{z}_n||^2_\lambda +b_1A^2|\nabla \tilde{u}_n|_2^2+b_2B^2|\nabla \tilde{v}_n|_2^2-\int_{\R^3}|\tilde{u}_n^+|^{\alpha}|\tilde{v}_n^+|^\beta dx.
	\end{aligned}
	\end{equation}
	Then by \eqref{24}, \eqref{211}--\eqref{2133} and \eqref{221}, we get that
	\begin{equation}
	\begin{aligned}
	o(1)=&||\tilde{z}_n||^2_\lambda-\int_{\R^3}|\tilde{u}_n^+|^{\alpha}|\tilde{v}_n^+|^\beta dx +b_1A^2|\nabla \tilde{u}_n|_2^2+b_2B^2|\nabla \tilde{v}_n|_2^2\\
	\geq& ||\tilde{z}_n||^2_\lambda-\int_{\R^3}|\tilde{u}_n|^{\alpha}|\tilde{v}_n|^\beta dx\\
	=&||\tilde{z}_n||^2_\lambda-\left(\int_{\R^3}|\tilde{u}_n|^{\alpha}|\tilde{v}_n|^\beta dx\right)^{\frac{\alpha+\beta-2}{\alpha+\beta}}
	\left(\int_{\R^3}|\tilde{u}_n|^{\alpha}|\tilde{v}_n|^\beta dx\right)^{\frac{2}{\alpha+\beta}}\\
	\geq &||\tilde{z}_n||^2_\lambda-(\hat{c})^{\frac{\alpha+\beta-2}{\alpha+\beta}}(2T)^{\alpha+\beta-2}\left(\frac{1}{2\lambda c}\right)^{\frac{1-\theta}{8\theta+1}}S^{-\frac{9\theta}{8\theta+1}}||\tilde{z}_n||^2_\lambda,
	\end{aligned}
	\end{equation}
	which implies that there exists $\Lambda^*> 1$ such that $\tilde{z}_n\to 0$ in $E_\lambda$ for all $\lambda>\Lambda^*$ and $0<b_1+b_2<b_*$. Consequently, this completes the proof.
\end{proof}
\begin{proof}[\rm{}\textbf{Proof of Theorem \ref{the1}}]
	Let $T=\sqrt{\frac{2(\alpha+\beta)(D+1)}{\alpha+\beta-2}}$.  By Lemmas \ref{L233}--\ref{L244}, we see that $\mathcal{J}_{\bf{b},\lambda}^T$ possesses the Mountain Pass geometry. Using a version of the Mountain Pass Theorem without (PS) condition \cite{mw}, we can obtain that for any $\lambda\geq 1$, and $0<b_1+b_2<b^*$, $\mathcal{J}_{\bf{b},\lambda}^T$ has a $\textup{(PS)}_{c^{T}_{\bf{b},\lambda}}$ sequence $\{z_n\}=\{(u_n,v_n)\}\subset E_\lambda$. By Lemmas \ref{L255}--\ref{L26}, we can get that there exists $b_*\in(0,b^*)$ such that for any $\lambda\geq 1$ and $b_1+b_2\in(0,b_*)$, up to a  subsequence, $\{z_n\}$ is a $\textup{(PS)}_{c^{T}_{\bf{b},\lambda}}$ sequence of $\mathcal{J}_{\bf{b},\lambda}$ satisfying
	\[\sup\limits_{n\in\mathbb{N}}||z_n||_{\lambda}\le T,\hspace{1ex}\mathcal{J}_{\bf{b},\lambda}(z_n)\to c^{T}_{\bf{b},\lambda}\hspace{1ex}\text{and}\hspace{1ex}||\mathcal{J}_{\bf{b},\lambda}'(z_n)||_{E^*_\lambda}\to 0,\hspace{1ex}\text{as}\hspace{1ex}n\to\infty.\]
	By Proposition \ref{L28}, there exists $\Lambda^*> 1$ such that  $\mathcal{J}_{\bf{b},\lambda}$ satisfies $\textup{(PS)}_{c^{T}_{\bf{b},\lambda}}$ condition in $E_\lambda$ for any $b_1+b_2\in(0,b_*)$ and $\lambda\in(\Lambda^*,\infty)$. Thus, we may assume that $z_n\to z_{\bf{b},\lambda}$ in $E_\lambda$, and then
	\[||z_{\bf{b},\lambda}||_{\lambda}\le T,\hspace{1ex}\mathcal{J}_{\bf{b},\lambda}(z_{\bf{b},\lambda})= c^{T}_{\bf{b},\lambda}\geq\eta>0\hspace{1ex}\text{and}\hspace{1ex}\mathcal{J}_{\bf{b},\lambda}'(z_{\bf{b},\lambda})= 0.\]
	Since 
	\[\begin{aligned}
	0=&\langle \mathcal{J}_{\bf{b},\lambda}'(z_{\bf{b},\lambda}),(u^-_{\bf{b},\lambda},v^-_{\bf{b},\lambda})\rangle\\
	=&||z^-_{\bf{b},\lambda}||^2_\lambda+b_1|\nabla u|^2_2\cdot|\nabla u^-|^2_2+b_2|\nabla v|^2_2\cdot|\nabla v^-|^2_2\\
	\geq&  ||z^-_{\bf{b},\lambda}||^2_\lambda,
	\end{aligned}\]
	we get that $u_{\bf{b},\lambda},v_{\bf{b},\lambda}\geq 0$ in $\R^3$. By using the strong maximum principle in each equation of \eqref{11}, we have that $u_{\bf{b},\lambda},v_{\bf{b},\lambda}> 0$ in $\R^3$. Moreover, note that $z_{\bf{b},\lambda}\neq \bf{0}$ and
	\begin{equation}\label{3133}
	\begin{aligned}
	0=&\langle \mathcal{J}_{\bf{b},\lambda}'(z_{\bf{b},\lambda}),z_{\bf{b},\lambda}\rangle\\
	=&||z_{\bf{b},\lambda}||^2_\lambda
	+b_1|\nabla u_{\bf{b},\lambda}|_2^4+b_2|\nabla v_{\bf{b},\lambda}|_2^4 -\int_{\R^3}|u_{\bf{b},\lambda}|^{\alpha}|u_{\bf{b},\lambda}|^\beta dx\\
	\geq &||z_{\bf{b},\lambda}||^2_\lambda-\hat{c}||z_{\bf{b},\lambda}||^{\alpha+\beta},
	\end{aligned}
	\end{equation}
	then, there exists $\delta >0$, independent of $b_1,b_2$ and $\lambda$, such that $||z_{\bf{b},\lambda}||_\lambda>\delta$ for any $b_1,b_2$ and $\lambda$. The proof is complete.
\end{proof}
\vskip2mm
{\section{Asymptotic behavior of positive vector solutions }\label{sec4}}
\setcounter{equation}{0}
\vskip2mm
In this section, we study the asymptotic behavior of positive vector solutions for \eqref{11} and and give the proofs of Theorem \ref{the2}, \ref{the3} and \ref{the4}.
\par
\vskip2mm
\begin{proof}[\rm{}\textbf{Proof of Theorem \ref{the2}}]
	For any sequence $\lambda_n\to\infty$, let $b_1+b_2\in(0,b_*)$ be fixed and $z_n:=z_{\textbf{b},\lambda_n}$ be the  positive vector  solution of $\mathcal{K}_{\textbf{b},\lambda_n}$ obtained in Theorem \ref{the1}. By \eqref{13}, we can get that for any $n\in\mathbb{N}$,
	\begin{equation}\label{31}
	0<\delta\le ||z_n||_{\lambda_n}\le T.
	\end{equation}
	Then there exist a subsequence $\{z_n\}=\{(u_n,v_n)\}$ and $z_{\bf{b}}=(u_{\bf{b}},v_{\bf{b}})$ in $E$ such that
	\begin{equation}\label{322}
	\begin{cases}
	(u_n,v_n)\rightharpoonup(u_{\bf{b}},v_{\bf{b}}),& \hspace{1ex}\textup{in} \hspace{1ex}E;\\
	(u_n,v_n)\to(u_{\bf{b}},v_{\bf{b}}),& \hspace{1ex}\textup{a.e. in} \hspace{1ex}\R^3;\\
	(u_n,v_n)\to(u_{\bf{b}},v_{\bf{b}}),& \hspace{1ex}\textup{in} \hspace{1ex}L_{loc}^{s_1}(\R^3)\times L_{loc}^{s_2}(\R^3),\hspace{1ex} 2\le s_1, s_2<6.\\
	\end{cases}
	\end{equation}
	We first prove $u_{\bf{b}}=0$ in $\Omega_1^c$. For $m\in\mathbb{N}$, set $C_m=\left\{ x\in\R^3: V(x)\geq\frac{1}{m} \right\}$, it follows from \eqref{31} that 
	\[\int_{C_m}u_n^2dx\le\frac{m}{\lambda_n}\int_{C_m}\lambda_nV(x)u_n^2dx\le\frac{m}{\lambda_n}||z_n||_{\lambda_n}^2\]
	and thus 
	\[\int_{C_m}u_{\bf{b}}^2dx\le \liminf\limits_{n\to \infty }\int_{C_m}u_n^2dx=0,\]
	which implies that $u_{\bf{b}}\equiv 0$ in $\Omega_1^c=\bigcup_{m=1}^{n} C_m$. Since $\Omega_1$ has smooth boundary, we get that $u_{\bf{b}}\in H_0^1(\Omega_1)$. Similarly, we have $v_{\bf{b}}\in H_0^1(\Omega_2)$.
	\par
	For any $\varphi\in C_0^\infty(\Omega_1)$, using $(\varphi,0)$ as a test function, we have
	\begin{equation}
	\left(1+b_1|\nabla u_n|^2_2\right)\int_{\R^3}\nabla u_n\nabla\varphi dx=\frac{\alpha}{\alpha+\beta}\int_{\R^3}|u_n^+|^{\alpha-2}u_n^+\varphi|v_n^+|^\beta dx,
	\end{equation}
	Since $\varphi$ has compact support, using the same argument as that in \eqref{212} and taking limit in the above equality, we get
	\begin{equation}\label{34}
	\left(1+b_1|\nabla u_n|^2_2\right)\int\limits_{\Omega_1\cup\Omega_2}\nabla u_{\bf{b}}\nabla\varphi dx=\frac{\alpha}{\alpha+\beta}\int\limits_{\Omega_1\cup\Omega_2}|u_{\bf{b}}^+|^{\alpha-2}u_{\bf{b}}^+\varphi|v_{\bf{b}}^+|^\beta dx,\hspace{1ex} \forall \varphi\in C_0^\infty(\Omega_1).
	\end{equation}
	Similarly, for any $\psi\in C_0^\infty(\Omega_2)$, we have 
	\begin{equation}\label{35}
	\left(1+b_2|\nabla v_n|^2_2\right)\int\limits_{\Omega_1\cup\Omega_2}\nabla v_{\bf{b}}\nabla\psi dx=\frac{\beta}{\alpha+\beta}\int\limits_{\Omega_1\cup\Omega_2}|u_{\bf{b}}^+|^{\alpha}|v_{\bf{b}}^+|^{\beta-2}v_{\bf{b}}^+\psi dx,\hspace{1ex} \forall \psi\in C_0^\infty(\Omega_2).
	\end{equation}
	\par
	Next we show that $z_n\to z_{\bf{b}}$ in $E$.
	Denote $\tilde{u}_n=u_n-u_{\bf{b}}$ and $\tilde{v}_n=v_n-v_{\bf{b}}$, then $\tilde{z}_n=z_n-z_{\bf{b}}$. Similiar to \eqref{2133}, we have 
	\begin{equation}\label{46}
	\begin{aligned}
	\int_{\R^3}|\tilde{u}_n|^\alpha|\tilde{v}_n|^\beta dx
	&\le S^{-\frac{9\theta}{2\theta+1}}||\tilde{z}_n||_{\lambda_n}^{\alpha+\beta-2+\frac{6\theta}{2\theta+1}}\left(\int_{\R^3}|\tilde{u}_n\tilde{v}_n|dx\right)^{\frac{1-\theta}{2\theta+1}}\\
	&\le \left(\frac{1}{2\lambda_n c}\right)^{\frac{1-\theta}{2\theta+1}}S^{-\frac{9\theta}{2\theta+1}}||\tilde{z}_n||_{\lambda_n}^{\alpha+\beta}+o(1)=o(1),
	\end{aligned}
	\end{equation}
	this together with Lemma \ref{BL} implies that
	\begin{equation}\label{38}
	\lim\limits_{n\to\infty}\int_{\R^3}|u_n^+|^{\alpha}|v_n^+|^{\beta} dx=\int_{\R^3}|u_{\bf{b}}^+|^{\alpha}|v_{\bf{b}}^+|^{\beta} dx=\int\limits_{\Omega_1\cup\Omega_2}|u_{\bf{b}}^+|^{\alpha}|v_{\bf{b}}^+|^{\beta} dx.
	\end{equation}
	By \eqref{212}--\eqref{213} and
	\[\langle \mathcal{J}_{b,\lambda_n}'(z_n),z_n\rangle=\langle \mathcal{J}_{b,\lambda_n}'(z_n),(u_{\bf{b}},0)\rangle=\langle \mathcal{J}_{b,\lambda_n}'(z_n),(0,v_{\bf{b}})\rangle=o(1),\]
	we have 
	\begin{equation}
	||z_n||_{\lambda_n}^2+b_1|\nabla u_n|^4_2+b_2|\nabla v_n|^4_2=\int_{\R^3}|u_n^+|^{\alpha}|v_n^+|^{\beta} dx+o(1),
	\end{equation}
	\begin{equation}
	||u_{\bf{b}}||^2+b_1|\nabla u_n|^2_2|\nabla u_{\bf{b}}|^2_2=\frac{\alpha}{\alpha+\beta}\int_{\R^3}|u_{\bf{b}}^+|^{\alpha}|v_{\bf{b}}^+|^{\beta} dx+o(1),
	\end{equation}
	\begin{equation}\label{311}
	||v_{\bf{b}}||^2+b_2|\nabla v_n|^2_2|\nabla v_{\bf{b}}|^2_2=\frac{\beta}{\alpha+\beta}\int_{\R^3}|u_{\bf{b}}^+|^{\alpha}|v_{\bf{b}}^+|^{\beta} dx+o(1).
	\end{equation}
	Up to a subsequence, we may assume that there exists $l_i>0(i=1,2,3)$ such that
	\[||z_n||_{\lambda_n}^2\to l_1,\hspace{1ex}|\nabla u_n|^2_2\to l_2,\hspace{1ex}|\nabla v_n|^2_2\to l_3.\]
	It follows from \eqref{38}--\eqref{311} that
	\[\begin{aligned}
	l_1+b_1l_2^2+b_2l_3^2&=||u_{\bf{b}}||^2+b_1l_2|\nabla u_{\bf{b}}|^2_2+||v_{\bf{b}}||^2+b_2l_3|\nabla v_{\bf{b}}|^2_2\\
	&\le ||z_{\bf{b}}||^2+b_1l_2^2+b_2l_3^2.
	\end{aligned}\]
	Then, $l_1\le ||z_{\bf{b}}||^2$. By the weakly lower semi-continuity of $||\cdot||$, we have
	\[||z_{\bf{b}}||^2\le \liminf\limits_{n\to\infty}||z_n||^2\le \limsup\limits_{n\to\infty}||z_n||^2\le \lim\limits_{n\to\infty}||z_n||^2_{\lambda_n}=l_1\le||z_{\bf{b}}||^2,\]
	which implies that $z_n\to z_{\bf{b}}$ in $E$.
	\par
	Finally, we are going to show that $z_{\bf{b}}$ is a positive vector solution of $\mathcal{K}_{\bf{b},\infty}$. It follows from \eqref{34}--\eqref{35} that for any $\varphi\in C_0^\infty(\Omega_1)$ and $\psi\in C_0^\infty(\Omega_2)$,
	\begin{equation}
	\left(1+b_1\int_{\Omega_1}|\nabla u_{\bf{b}}|^2 dx\right)\int_{\Omega_1}\nabla u_{\bf{b}}\nabla\varphi dx=\frac{\alpha}{\alpha+\beta}\int_{\Omega_1}|u_{\bf{b}}^+|^{\alpha-2}u_{\bf{b}}^+\varphi|v_{\bf{b}}^+|^\beta dx
	\end{equation}
	and
	\begin{equation}
	\left(1+b_2\int_{\Omega_2}|\nabla v_{\bf{b}}|^2 dx\right)\int_{\Omega_2}\nabla v_{\bf{b}}\nabla\psi dx=\frac{\beta}{\alpha+\beta}\int_{\Omega_2}|u_{\bf{b}}^+|^{\alpha}|v_{\bf{b}}^+|^{\beta-2}v_{\bf{b}}^+\psi dx.
	\end{equation}
	Thus, $z_{\bf{b}}=(u_{\bf{b}},v_{\bf{b}})$ is a nonnegative vector solution of $\mathcal{K}_{\bf{b},\infty}$ by the density of $C_0^\infty(\Omega_i)$ in $H_0^1(\Omega_i)(i=1,2)$. By \eqref{31} and $z_n\to z_{\bf{b}}$ in $E$, we have
	\[||z_{\bf{b}}||=\lim\limits_{n\to\infty}||z_n||_{\lambda_n}\geq \delta>0,\]
	and so $u_{\bf{b}},v_{\bf{b}}\neq0$. 
	We can deduce that $u_{\bf{b}}>0$ in $\Omega_1$ and $v_{\bf{b}}>0$ in $\Omega_2$ by the strong maximum principle. The proof is complete.
\end{proof}
\begin{proof}[\rm{}\textbf{Proof of Theorem \ref{the3}}]
	For any sequence $b_n\to0$, let $\lambda\in(\Lambda^*,\infty)$ be fixed and $z_n:=z_{\textbf{b}_n,\lambda}$ be the  positive vector  solution of $\mathcal{K}_{\textbf{b}_n,\lambda}$ obtained in Theorem \ref{the1}. By \eqref{13}, we can get that for any $n\in\mathbb{N}$,
	\begin{equation}\label{313}
	0<\delta\le ||z_n||_{\lambda}\le T,
	\end{equation}
	i.e., $\{z_n\}$ is bounded in $E_\lambda$. Then there exist a subsequence $\{z_n\}$ and $z_\lambda=(u_\lambda,v_\lambda)$ in $E_\lambda$ such that $z_n\rightharpoonup z_\lambda$ in $E_\lambda$. By a similar argument as used in the proof of Proposition \ref{L28}, we get that $z_n\to z_\lambda$ in $E_\lambda$.
	\par
	Now, it is enough to show that $z_\lambda$ is a positive vector solution of \ref{3res}. Note that $\mathcal{J}_{b_n,\lambda}'(z_n)\to0$, then for any $\zeta=(\varphi,\psi)\in E_\lambda$, using $(\varphi,0)$ as a test function, we have
	\begin{equation}\label{314}
	\int_{\R^3}(\nabla u_\lambda\nabla\varphi+\lambda V(x)u_\lambda\varphi) dx=\frac{\alpha}{\alpha+\beta}\int_{\R^3}|u_\lambda^+|^{\alpha-2}u_\lambda^+\varphi|v_\lambda^+|^\beta dx.
	\end{equation}
	Similarly, using $(0,\psi)$ as a test function, we have
	\begin{equation}\label{315}
	\int_{\R^3}(\nabla v_\lambda\nabla\psi+\lambda W(x)v_\lambda\psi) dx=\frac{\beta}{\alpha+\beta}\int_{\R^3}|u_n^+|^{\alpha}|v_n^+|^{\beta-2}v_\lambda^+\psi dx.
	\end{equation}
	Thus, $z_\lambda=(u_\lambda,v_\lambda)$ is a nonnegative vector solution of \ref{3res}. By \eqref{313} and $z_n\to z_\lambda$ in $E_\lambda$, we have
	\[||z_\lambda||_\lambda=\lim\limits_{n\to\infty}||z_n||_{\lambda}\geq \delta>0,\]
	and so $u_\lambda,v_\lambda\neq0$.
	We can deduce that $u_\lambda>0$ in $\R^3$ and $v_\lambda>0$ in $\R^3$ by the strong maximum principle. The proof is complete.
\end{proof}
\begin{proof}[\rm{}\textbf{Proof of Theorem \ref{the4}}] The proof of Theorem \ref{the4} is essentially same as the proof of Theorem \ref{the2}, so we omit it here.
\end{proof}







\par\noindent
\textbf{Acknowledgments}
\vskip2mm
This work is  supported by National Natural Science Foundation of China (No. 12071486).













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