\section{Introduction}
Let $N\geq 2$, $s\in (0, 1)$ and $p\in (1, \frac{N}{s})$.
Let us consider the following nonlinear problem
\begin{equation}\label{P}
\left\{
\begin{array}{ll}
(-\Delta)^{s}_{p} u= g(u) \, \mbox{ in } \mathbb{R}^{N}, \\
u\in W^{s, p}(\mathbb{R}^{N}), 
\end{array}
\right.
\end{equation}
where $(-\Delta)^{s}_{p}$ is the fractional $p$-Laplacian operator (see \cite{CL, DTGCV, IMS, P}) defined by
$$
(-\Delta)^{s}_{p}u(x)=C_{N, s, p} P.V. \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}dy, \quad 
C_{N, s, p}=\frac{\frac{sp}{2}(1-s)2^{2s-1}}{\pi^{\frac{N-1}{2}}} \frac{\Gamma(\frac{N+sp}{2})}{\Gamma(\frac{p+1}{2})}\Gamma(2-s),
$$
for all $u\in C^{1, 1}_{loc}(\mathbb{R}^{N})\cap L^{p-1}_{sp}$, with
$$
L^{p-1}_{sp}=\Bigl\{u\in L^{p-1}_{loc}(\mathbb{R}^{N}): \int_{\mathbb{R}^{N}}\frac{|u(x)|^{p-1}}{1+|x|^{N+sp}}dx<\infty\Bigr\},
$$
and
$g:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous nonlinearity such that $g(0)=0$, $g$ is odd and satisfies the following Berestycki-Lions type conditions \cite{BL}:
\begin{compactenum}[$(g1)$]
\item $\displaystyle{-\infty<\liminf_{t\rightarrow 0} \frac{g(t)}{|t|^{p-2}t}\leq \limsup_{t\rightarrow 0} \frac{g(t)}{|t|^{p-2}t}=-m}$, with $m>0$,
\item $\displaystyle{-\infty\leq \limsup_{t\rightarrow 0} \frac{g(t)}{|t|^{p^{*}_{s}-1}}\leq 0}$, where $p^{*}_{s}=\frac{Np}{N-sp}$ is the critical fractional exponent,
\item there exists $\zeta>0$ such that $G(\zeta)>0$, where $G(t)=\int_{0}^{t}g(\tau)d\tau$.
\end{compactenum}
Our first result deals with the existence of a weak solution to \eqref{P} satisfying a Pohozaev type identity. More precisely, we prove the following result. 
\begin{thm}\label{thm1}
Let $N\geq 2$, $s\in (0, 1)$ and $p\in (1, \frac{N}{s})$. Assume that $(g1)$-$(g3)$ hold. Then there exists a nontrivial weak solution $u\in W^{s, p}(\mathbb{R}^{N})$ to \eqref{P} fulfilling the following Pohozaev identity: 
\begin{align}\label{POHOZAEV}
\frac{C_{N, s, p}(N-sp)}{2p}\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy-N\int_{\mathbb{R}^{N}} G(u)dx=0.
\end{align}
\end{thm}
We recall that the Pohozaev identity for the $p$-Laplacian operator $\Delta_{p}u=\dive(|\nabla u|^{p-2}\nabla u)$ has been extensively investigated in literature; see for instance \cite{GV, PS}.
On the other hand, in the fractional setting, if $s\in (0, 1)$ and $p=2$, then $(-\Delta)^{s}_{p}$ becomes the fractional Laplacian operator $(-\Delta)^{s}$ and the corresponding Pohozaev identity for weak solutions to \eqref{P} has been established in \cite[Proposition 4.1]{CW} (see also \cite[Theorem 3.5.1]{A} for more details and \cite{RS} for the case of bounded domains). More precisely, in \cite{CW} the authors required $g\in C^{1}(\mathbb{R})$ and obtained the Pohozaev identity for $(-\Delta)^{s}$ by combining the extension method \cite{CS}, some regularity results for $(-\Delta)^{s}$, 
an integration by parts and some ideas inspired by the proof given in \cite[Proposition 1]{BL}.  
When $p\neq 2$, we are not able to adapt this approach. In fact, even if the extension method for $(-\Delta)^{s}_{p}$ has been recently studied in \cite{DTGCV}, we do not have enough informations about the regularity of the extension $U(x, y)$ of $u(x)$ in $\mathbb{R}^{N+1}_{+}=\{(x, y)\in \mathbb{R}^{N+1}: y>0 \}$ whenever $p\neq 2$, and contrary to the linear case $p=2$ we do not know if the following identity is valid:
$$
C\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy=\iint_{\mathbb{R}^{N+1}_{+}} y^{-1+p(1-s)} |\nabla U|^{p}dxdy,
$$
where $C>0$ is an appropriate constant. 
Therefore, we need to follow a different strategy to arrive at \eqref{POHOZAEV}. We stress that the main difficulty in reaching \eqref{POHOZAEV}, it consists in using $x\cdot \nabla u$, where $u$ is a weak solution to \eqref{P}, as test function in the weak formulation of \eqref{P} and then apply an integration by parts. Indeed, due to the fact that for $p\neq 2$ we have to handle the nonlinearity of the operator $(-\Delta)^{s}_{p}$ and its nonlocal character, it seems to be an awkward task to verify that $x\cdot \nabla u\in W^{s, p}(\mathbb{R}^{N})$. Moreover, an integration by parts formula for $(-\Delta)^{s}_{p}$ seems to be hard to accomplish.
In this paper, by means of suitable variational methods for potential operators with covariance condition (see \cite{B, Chab}), we prove the existence of a weak solution to \eqref{P} and that satisfies \eqref{POHOZAEV}.
We recall that $u\in W^{s, p}(\mathbb{R}^{N})$ is a weak solution to \eqref{P} if  $\langle I'(u),\phi\rangle=0$ for all $\phi\in W^{s, p}(\mathbb{R}^{N})$, where $I:W^{s, p}(\mathbb{R}^{N})\rightarrow \mathbb{R}$ is  the energy functional associated with \eqref{P}, namely
$$
I(u)=\frac{C_{N, s, p}}{2p}\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy-\int_{\mathbb{R}^{N}} G(u)dx.
$$
Note that the restriction $N\geq 2$ ensures that \eqref{P} admits a radial weak solution (see Theorem \ref{THMBL}).
We stress that Theorem \ref{thm1} only guarantees the existence of a weak solution to \eqref{P} fulfilling \eqref{POHOZAEV} and not that, as in the case $p=2$, any weak solution to \eqref{P} satisfies \eqref{POHOZAEV}.
However, this last statement is true for $C^{1,1}$ functions satisfying equation \eqref{P} in the pointwise sense.
More precisely, our second main result can be stated as follows. 
\begin{thm}\label{thm2}
Let $N\geq 2$, $s\in (0, 1)$ and $p\in (1, \frac{N}{s})$.  
When $p\in (1, 2)$, we assume that $s<\frac{2(p-1)}{p}$. Assume that $(g1)$-$(g3)$ hold.
Let $u\in W^{s, p}(\mathbb{R}^{N})\cap C^{1, 1}(\mathbb{R}^{N})$ be such that $(-\Delta)^{s}_{p}u=g(u)$ in $\mathbb{R}^{N}$. Then \eqref{POHOZAEV} is valid.
\end{thm}
The proof of Theorem \ref{thm2} goes as follows. We first prove an integral representation formula for $(-\Delta)^{s}_{p}v$, with $v\in C^{1, 1}(\mathbb{R}^{N})$, in the spirit of \cite[Lemma 3.2]{DPV} (see Lemma \ref{RIF}). Thus, motivated by \cite[Lemma 4.2]{DFW1}, we establish an integration by parts formula for $C^{1, 1}$ functions and vector fields of class $C^{0, 1}_{c}$ (see Lemma \ref{LEMMA}). 
Therefore, multiplying \eqref{P} by $\varphi(\lambda x) x\cdot \nabla u$, 
where $\varphi\in C^{1}_{c}(\mathbb{R}^{N})$ is such that $\varphi=1$ in a neighborhood of $0$ and $\lambda>0$,  integrating over $\mathbb{R}^N$ and using our integration by parts formula, we can pass to the limit as $\lambda\rightarrow 0$ in the relation that we find to infer the desired result.


As far as we know, this is the first time that a Pohozaev identity for the fractional $p$-Laplacian operator $(-\Delta)^{s}_{p}$ in $\mathbb{R}^{N}$ is obtained in literature. \\
The paper is organized as follows. In section $2$ we give the proof of Theorem \ref{thm1}. Section $3$ is devoted to the proof of Theorem \ref{thm2}.

\subsection*{Notations:}
Henceforth, $\|\cdot\|_{L^{p}(\mathbb{R}^{N})}$ denotes the $L^{p}(\mathbb{R}^N)$-norm with $p\in [1, \infty]$, for $s\in (0, 1)$ and $p\in [1, \infty)$ the fractional Sobolev space 
$W^{s, p}(\mathbb{R}^{N})$ is the set of functions $u\in L^{p}(\mathbb{R}^{N})$ such that $\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy<\infty$,
$D^{\beta} u:=\frac{\partial^{|\beta|}u}{{\partial x_{1}^{\beta_{1}}}\dots {\partial x_{N}^{\beta_{N}}}}$ stands for the distributional derivative of order $\beta=(\beta_{1}, \dots, \beta_{N})\in (\mathbb{N}\cup\{0\})^{N}$ ($\beta$ is called a multi-index), where $|\beta|:=\beta_{1}+\dots+\beta_{N}$ is the order of $\beta$. With $C^{k, \gamma}(\mathbb{R}^{N})$, where $k\in \mathbb{N}\cup \{0\}$ and $\gamma\in (0, 1]$, we denote the set of functions $u\in C^{k}(\mathbb{R}^N)$ whose derivatives up to order $k$ are bounded and such that
$$
 \|u\|_{C^{k, \gamma}(\mathbb{R}^{N})}=\sum_{|\beta|\leq k}\|D^{\beta}u\|_{L^{\infty}(\mathbb{R}^{N})}+\sum_{|\beta|=k}\sup_{x\neq y} \frac{|D^{\beta}u(x)-D^{\beta}u(y)|}{|x-y|^{\gamma}}<\infty. 
$$ 



\section{Proof of Theorem \ref{thm1}}
In this section, we investigate the existence of a weak solution to \eqref{P} by means of two abstract results found in \cite[Chapter 4]{Chab} (see also \cite{B}). For the reader's convenience, we state them below. We first introduce some definitions.

Let $X$ be a reflexive Banach space equipped with norm $\|\cdot\|$ and denote by $X^{*}$ its dual. 
A map $A:X\rightarrow X^{*}$ is said to be a potential operator with a potential $a:X\rightarrow \mathbb{R}$, if $a$ is Gateaux differentiable and
$$
\lim_{t\rightarrow 0} \frac{a(u+tv)-a(u)}{t}=\langle A(u), v\rangle \quad \mbox{ for all } u, v\in X.
$$
For a potential, we alway assume that $a(0)=0$. 
Let $A:X\rightarrow X^{*}$ and $B:X\rightarrow X^{*}$ two potential operators with potentials $a:X\rightarrow \mathbb{R}$ and $b:X\rightarrow \mathbb{R}$, respectively.
Suppose that $b$ is defined on a set $D(b)\subset X$.
For all $\sigma>0$, let $T_{\sigma}: X\rightarrow X$ be a linear map such that $T_{\sigma_1}\circ T_{\sigma_{2}}=T_{\sigma_{1}\sigma_{2}}$ for all $\sigma_{1}, \sigma_{2}>0$, and $T_{1}=id$. Let $E$ be a subspace of $X$ such that $T_{\sigma}(E)\subset E$ for each $\sigma>0$. We assume the following covariance conditions on the potentials $a$ and $b$:
\begin{compactenum}[$(C1)$]
\item $a: X\rightarrow \mathbb{R}$ and there exists $q_{1}\in \mathbb{R}$ such that $a(T_{\sigma}u)=\sigma^{q_{1}}a(u)$ for each $\sigma>0$ and $u\in D(b)$.
\item $b: D(b)\rightarrow \mathbb{R}$ with $E\subset D(b)$, $D(b)+E\subset D(b)$, $T_{\sigma}(D(b))\subset D(b)$, and there exists $q_{2}\in \mathbb{R}$ such that $b(T_{\sigma}u)=\sigma^{q_{2}}b(u)$ for each $\sigma>0$ and $u\in D(b)$.
\end{compactenum}
Assuming $\{u\in D(b): b(u)=1\}\neq \emptyset$, we consider
\begin{align}\label{J1}
J=\inf\{a(u): u\in D(b), \, b(u)=1 \}.
\end{align}
\begin{thm}\label{THM4.1.1}\cite[Theorem 4.1.1-(i)]{Chab}
Suppose that $A$ is a potential operator with a potential $a$. Moreover, suppose that the problem \eqref{J1} has a solution $u$ and that $b$ has a linear continuous Gateaux derivative $\langle B(u), v\rangle$ at all directions $v\in E$. Then
$$
\langle A(u), v\rangle=J \frac{q_{1}}{q_{2}}\langle B(u), v\rangle \quad \mbox{ for all } v\in E.
$$
The functionals $a$ and $b$ have linear continuous Gateaux derivatives at all directions $T_{\sigma}u$, $\sigma>0$, satisfying 
$$
\langle A(T_{\sigma}u), v\rangle=\frac{q_{1}}{q_{2}}J \sigma^{q_{1}-q_{2}}\langle B(T_{\sigma}u), v\rangle \quad \mbox{ for all } v\in E \mbox{ and } \sigma>0.
$$
Furthermore, if $\frac{q_{1}}{q_{2}}J>0$ and $q_{1}-q_{2}\neq 0$, then the scaled minimizer $\bar{u}=T_{\bar{\sigma}}u$, with $\bar{\sigma}=\left(\frac{q_{1}}{q_{2}}J\right)^{\frac{1}{q_{2}-q_{1}}}$, satisfies
$$
\langle A(\bar{u}), v\rangle=\langle B(\bar{u}), v\rangle \quad \mbox{ for all } v\in E.
$$
\end{thm}
Next we assume that $a$ and $b$ satisfy $(C1)$ and $(C2)$ with $D(b)=X$ and $q_{1}\neq q_{2}$.
\begin{thm}\label{THM4.1.2}\cite[Theorem 4.1.2]{Chab}
Suppose that there exists a Banach subspace $X_{1}\subset X$ equipped with a norm from $X$ and a mapping $T:X\rightarrow X_{1}$ such that $a(Tu)\leq a(u)$ and $b(Tu)=b(u)$ for each $u\in X$ and let $b(u_{0})>0$ for some $u_{0}\in X$. Moreover, we assume that $a$ is weakly lower semicontinuous and that $b$ admits a decomposition $b(u)=b_{1}(u)-b_{2}(u)$ on $X$, with $b_{1}$ and $b_{2}$ nonnegative on $X$, where $b_{1}$ restricted on $X_{1}$ is weakly continuous, $b_{2}$ is weakly lower semicontinuous and 
\begin{compactenum}[$(i)$]
\item there exist constants $\e\in (0, 1)$, $C>0$ and $\alpha>0$ such that 
$$
b_{1}(u)\leq \e b_{2}(u)+Ca(u)^{\alpha} \quad \mbox{ for all } u\in X,
$$
\item there exist constants $\beta>0$, $C'>0$ and $\gamma>0$ such that 
$$
\|u\|^{\beta}\leq C'(b_{2}(u)+a(u)^{\gamma}) \quad \mbox{ for all } u\in X.
$$
\end{compactenum}
Then problem \eqref{J1} has a nontrivial solution.
\end{thm}

\noindent
Now we are able to prove an existence result for \eqref{P}.
\begin{thm}\label{THMBL}
Let $N\geq 2$, $s\in (0, 1)$ and $p\in (1, \frac{N}{s})$.
Then, \eqref{P} admits a nontrivial weak solution.
\end{thm}
\begin{proof}
Set $X=W^{s,p}(\mathbb{R}^{N})$ and $X_{1}=W^{s, p}_{rad}(\mathbb{R}^{N})=\{u\in W^{s, p}(\mathbb{R}^{N}): u \mbox{ is radially symmetric } \}$. It is easy to check that $X$ is a reflexive Banach space. In fact, let $Y=L^{p}(\mathbb{R}^{N})\times L^{p}(\mathbb{R}^{2N})$ and  introduce the linear isometry $\mathcal{L} :W^{s, p}(\mathbb{R}^{N})\rightarrow Y$ defined as 
$$
\mathcal{L}(u)=\left(u, \frac{u(x)-u(y)}{|x-y|^{\frac{N+sp}{p}}} \right).
$$ 
Clearly, the product space $Y$ is reflexive.
Since $W^{s, p}(\mathbb{R}^{N})$ is a Banach space, $\mathcal{L}(W^{s, p}(\mathbb{R}^{N}))$ is a closed subspace of $Y$. It follows that $\mathcal{L}(W^{s, p}(\mathbb{R}^{N}))$ is reflexive. As a result, $W^{s, p}(\mathbb{R}^{N})$ is a reflexive space for all $s\in (0, 1)$ and $p\in (1, \infty)$.

Set
$$
\|u\|=\left(\frac{C_{N, s, p}}{2p} \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy+\|u\|^{p}_{L^{p}(\mathbb{R}^{N})}\right)^{\frac{1}{p}} \quad \mbox{ for all } u\in X.
$$
Let $T_{\sigma}: X\rightarrow X$ defined as $T_{\sigma}u(x)=u(\frac{x}{\sigma})$, $\sigma>0$, and $T:X\rightarrow X_{1}$ given by $Tu(x)=u^{*}(|x|)$ (Schwarz symmetrization). 
Modifying $g$ as in \cite{BL}, we may assume that $g$ satisfies the stronger condition
\begin{align}\label{G2'}
\lim_{|t|\rightarrow \infty} \frac{|g(t)|}{|t|^{p^{*}_{s}-1}}=0, 
\end{align}
instead of $(g2)$.
Put 
$$
a(u)=\frac{C_{N, s, p}}{2p}\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy \,\,\mbox{ and } \,\, b(u)=\int_{\mathbb{R}^{N}} G(u)dx.
$$ 
Note that $a, b\in C^{1}(X, \mathbb{R})$, $a(T_{\sigma}u)=\sigma^{N-sp} a(u)$ and $b(T_{\sigma}u)=\sigma^{N}b(u)$ for each $u\in X$ and $\sigma>0$. We also have
$a(Tu)\leq a(u)$ (see \cite[Theorem A.1]{FS}) and $b(Tu)=b(u)$ (see \cite[Appendix]{BL}) for all $u\in X$. Clearly, $a$ is weakly lower semicontinuous on $X$.

Let us now show that the set
$\{u\in X: b(u)=1\}$ is nonempty.
For $R>0$, define
\begin{equation*}
w_{R}(x)=\left\{
\begin{array}{ll}
\zeta & \mbox{ for } |x|\leq R, \\
\zeta(R+1-|x|) & \mbox{ for } R\leq |x|\leq R+1, \\
0 & \mbox{ for } |x|\geq R+1,
\end{array}
\right.
\end{equation*}
where $\zeta>0$ is given in $(g3)$.
It is clear that $w_{R}\in X$ and 
$$
b(w_{R})\geq G(\zeta)|B_{R}(0)|-|B_{R+1}(0)\setminus B_{R}(0)| \max_{t\in [0, \zeta]} |G(t)|.
$$
Then there exist two constants $C_{1}, C_{2}>0$ such that
$$
b(w_{R})\geq C_{1}R^{N}-C_{2}R^{N-1},
$$
and so $b(w_{R})>0$ for $R>0$ large enough. 

Next we consider the following constrained minimization problem:
\begin{align}\label{J}
J=\inf\{a(u): u\in X, \, b(u)=1\}.
\end{align}
Let $g_{1}(t)=(g(t)+mt^{p-1})^{+}$ and $g_{2}(t)=g_{1}(t)-g(t)$ for all $t\geq 0$. We extend $g_{1}$ and $g_{2}$ as odd functions for $t\leq 0$. Set $b_{1}(u)=\int_{\mathbb{R}^{N}} G_{1}(u)dx$ and $b_{2}(u)=\int_{\mathbb{R}^{N}}G_{2}(u)dx$, where $G_{i}(t)=\int_{0}^{t}g_{i}(\tau)d\tau$ for $i=1, 2$. Then, $b(u)=b_{1}(u)-b_{2}(u)$ for all $u\in X$. Clearly, $b_{1}(u), b_{2}(u)\geq 0$ for all $u\in X$, and $b_{2}$ is weakly lower semicontinuous on $X$. Let us prove that $b_{1}$ is weakly continuous on $X_{1}$. Let $(u_{n})\subset X_{1}$ be such that $u_{n}\rightharpoonup u$ in $X_{1}$ and we show that $b_{1}(u_{n})\rightarrow b_{1}(u)$. By \cite[Theorem II.1]{Lions}, we know that $X_{1}$ is compactly embedded into $L^{r}(\mathbb{R}^{N})$ for all $r\in (p, p^{*}_{s})$. We note that $G_{1}\in C(\mathbb{R})$, $\lim_{|t|\rightarrow 0} \frac{G_{1}(t)}{|t|^{p}}=0=\lim_{|t|\rightarrow \infty} \frac{G_{1}(t)}{|t|^{p^{*}_{s}}}$ (by definition of $g_{1}$ and assumptions $(g1)$ and \eqref{G2'}), $\sup_{n\in \mathbb{N}}\|u_{n}\|\leq C$ and $G(u_{n})\rightarrow G(u)$ a.e in $\mathbb{R}^{N}$. Applying \cite[Lemma 2.4]{CW}, we deduce that $\|G(u_{n})-G(u)\|_{L^{1}(\mathbb{R}^{N})}\rightarrow 0$ and thus $b_{1}(u_{n})\rightarrow b_{1}(u)$ as desired.

Hereafter, we verify that assumptions $(i)$ and $(ii)$ of Theorem \ref{THM4.1.2} hold. 
Using $(g1)$ and $(g2)$, we see that for all $\e>0$ there exists a constant $C_{\e}>0$ such that 
$$
G_{1}(t)\leq \e G_{2}(t)+C_{\e}|t|^{p^{*}_{s}} \quad \mbox{ for all } t\in \mathbb{R},
$$
and thanks to the Sobolev inequality (see \cite[Theorem 6.5]{DPV}) we arrive at 
$$
b_{1}(u)\leq \e b_{2}(u)+C_{\e} C_{*} a(u)^{\frac{p^{*}_{s}}{p}} \quad \mbox{ for all } u\in X,
$$
for some constant $C_{*}>0$.
On the other hand, because $g_{2}(t)\geq m t^{p-1}$ for all $t\geq 0$, we have
$$
b_{2}(u)\geq \frac{m}{p}\|u\|^{p}_{L^{p}(\mathbb{R}^{N})} \quad \mbox{ for all } u\in X,
$$
whence
$$
\|u\|^{p}\leq \frac{p}{m} b_{2}(u)+a(u) \quad \mbox{ for all } u\in X.
$$
Then, invoking Theorem \ref{THM4.1.2}, we infer that problem \eqref{J} has a nontrivial solution $u$. Observing that $\frac{(N-sp)}{N}J>0$ and $(N-sp)-N=-sp\neq 0$, we can invoke Theorem \ref{THM4.1.1} to deduce that $\bar{u}=T_{\bar{\sigma}}u$, where $\bar{\sigma}=(\frac{N-sp}{N}J)^{\frac{1}{sp}}$, satisfies
\begin{align*}
\langle a'(\bar{u}), v\rangle=\langle b'(\bar{u}), v\rangle \quad \mbox{ for all } v\in X,
\end{align*}
that is $\bar{u}$ is a weak solution to \eqref{P}. 
\end{proof}

As byproduct of Theorem \ref{THMBL}, we obtain Theorem \ref{thm1}.
\begin{proof}[Proof of Theorem \ref{thm1}]
Let $\bar{u}$ be the solution found in Theorem \ref{THMBL}.
From $a(\bar{u})=\bar{\sigma}^{N-sp}J$, $J>0$, $b(\bar{u})=\bar{\sigma}^{N}$ and $\bar{\sigma}=(\frac{N-sp}{N}J)^{\frac{1}{sp}}$, we can see that 
\begin{align*}
a(\bar{u})=\frac{N}{N-sp}b(\bar{u}),
\end{align*}
that is, \eqref{POHOZAEV} holds.
\end{proof}


\begin{remark}
The proof of Theorem \ref{thm1} still works in the case $s=1$.
\end{remark}

\begin{remark}
Reasoning as in \cite[Lemma 3.18]{AIDCDS}, we can show that if $u$ is a nonnegative weak solution to \eqref{P}, then $u\in C^{0, \alpha}(\mathbb{R}^{N})\cap L^{r}(\mathbb{R}^{N})$ for all $r\in [p, \infty]$ and for some $\alpha\in (0, 1)$. Since $(-\Delta)^{s}_{p}u+m u^{p-1}\leq g_{1}(u)$ in $\mathbb{R}^{N}$, where $g_{1}(t)=o(t^{p-1})$ as $t\rightarrow 0$, we can proceed as in  \cite[Theorem 1.5]{DPQ} to see that $|u(x)|\leq C|x|^{-\frac{N+sp}{p-1}}$ for all $|x|$ large enough. If in addition $u\in C^{1, 1}_{loc}(\mathbb{R}^{N})$ and $g\in C^{1}(\mathbb{R})$, then $u$ must be radially symmetric and monotone decreasing about some point in $\mathbb{R}^N$ in view of  \cite[Theorem 5]{CL}.
\end{remark}



\section{Proof of Theorem \ref{thm2}}



We start by giving a useful integral representation formula for $(-\Delta)^{s}_{p}$ along functions of class $C^{1, 1}(\mathbb{R}^{N})$.
This formula is motivated by \cite[Lemma 3.2]{DPV} in which the authors dealt with $p=2$ and functions in the Schwartz space of rapidly decaying functions. 
\begin{lem}\label{RIF}
Let $N\in \mathbb{N}$, $s\in (0, 1)$ and $p\in (1, \infty)$. When $p\in (1, 2)$, we assume that $s<\frac{2(p-1)}{p}$. Then, for all $u\in C^{1, 1}(\mathbb{R}^{N})$, it holds, for all $x\in \mathbb{R}^N$,
\begin{align*}
(-\Delta)^{s}_{p}u(x)=\frac{C_{N, s, p}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))+|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}} dz.
\end{align*}
\end{lem}
\begin{proof}
Pick $u\in C^{1, 1}(\mathbb{R}^{N})$ and fix $x\in \mathbb{R}^{N}$. Using the change of variables theorem and the symmetry of the kernel $\frac{1}{|z|^{N+sp}}$, we have 
\begin{align*}
&P.V.\int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}} dy\\
&\quad =P.V.\int_{\mathbb{R}^{N}} \frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))}{|z|^{N+sp}} dz \\
&\quad =\frac{1}{2} P.V.\int_{\mathbb{R}^{N}} \frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))+|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}} dz.
\end{align*}
Let us now show that the $P.V.$ in the above formula can be removed. 
Fix $\e\in (0, 1)$. Then we can write 
\begin{align*}
&\int_{\mathbb{R}^{N}\setminus B_{\e}(0)} \frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))+|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}} dz\\
&\quad =\int_{\mathbb{R}^{N}\setminus B_{1}(0)} \frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))+|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}} dz\\
&\quad \quad +\int_{B_{1}(0)\setminus B_{\e}(0)} \frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))+|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}} dz.
\end{align*}
We first prove that 
$$
\int_{\mathbb{R}^{N}\setminus B_{1}(0)} \frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))+|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}}dz<\infty.
$$
Take $|z|\geq 1$. Since $u\in L^{\infty}(\mathbb{R}^{N})$, we see that 
\begin{align*}
\left|\frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))+|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}}\right|\leq \frac{2^{p}\|u\|^{p-1}_{L^{\infty}(\mathbb{R}^{N})}}{|z|^{N+sp}}.
\end{align*}
Because $\frac{1}{|z|^{N+sp}}$ is integrable at infinity, the assertion is proved.
Now we claim that
$$
\int_{B_{1}(0)\setminus B_{\e}(0)} \frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))+|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}}dz<\infty.
$$
Let $\e\leq |z|<1$. We first assume $p\geq 2$.
Recalling that (see \cite[pag. 255]{Simon})
$$
||a|^{p-2}a-|b|^{p-2}b|\leq C_{p}(|a|+|b|)^{p-2}|a-b| \quad \mbox{ for all } a, b\in \mathbb{R},
$$
and using the mean value theorem, we obtain
\begin{align*}
&\left|\frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))+|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}}\right|\\
&\quad =\left|\frac{|\nabla u(x+\theta_{1}z)\cdot z|^{p-2}\nabla u(x+\theta_{1}z)\cdot z-|\nabla u(x-\theta_{2}z)\cdot z|^{p-2}\nabla u(x-\theta_{2}z)\cdot z}{|z|^{N+sp}}\right|\\
&\quad \leq 2^{p-2}C_{p} \|\nabla u\|^{p-2}_{L^{\infty}(\mathbb{R}^{N})}|z|^{p-2} \frac{|[\nabla u(x+\theta_{1}z)-\nabla u(x-\theta_{2}z)]\cdot z|}{|z|^{N+sp}} \\
&\quad \leq 2^{p-2}C_{p} \|\nabla u\|^{p-2}_{L^{\infty}(\mathbb{R}^{N})} |z|^{p-2} \frac{[\nabla u]_{C^{0, 1}(\mathbb{R}^{N})}|z|^{2}}{|z|^{N+sp}} \\
&\quad \leq 2^{p-2} C_{p} \|u\|_{C^{1, 1}(\mathbb{R}^{N})}^{p-1}\frac{1}{|z|^{N+sp-p}},
\end{align*}
where $\theta_{1}, \theta_{2}\in (0, 1)$. Since 
$$
\frac{1}{|z|^{N+sp-p}}
$$ 
is integrable near the origin (thanks to $s\in (0, 1)$), we reach the assertion. Now we suppose $p\in (1, 2)$. Then, observing that (see \cite[pag. 255]{Simon})
$$
||a|^{p-2}a-|b|^{p-2}b|\leq C'_{p}|a-b|^{p-1} \quad \mbox{ for all } a, b\in \mathbb{R},
$$
we get
\begin{align*}
&\left|\frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))+|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}}\right|\\
&\quad \leq \frac{|\nabla u(x+\theta_{1}z)\cdot z-\nabla u(x-\theta_{2}z)\cdot z|^{p-1}}{|z|^{N+sp}}\\
&\quad \leq C'_{p} \|\nabla u\|^{p-1}_{C^{0, 1}(\mathbb{R}^{N})} |z|^{p-1} \frac{|z|^{p-1}}{|z|^{N+sp}} \\
&\quad \leq C'_{p} \|u\|_{C^{1, 1}(\mathbb{R}^{N})}^{p-1}\frac{1}{|z|^{N+sp-2p+2}},
\end{align*}
where $\theta_{1}, \theta_{2}\in (0, 1)$. Due to the fact that 
$$
\frac{1}{|z|^{N+sp-2p+2}}
$$ 
is integrable near the origin (because of $s<\frac{2(p-1)}{p}$), we can conclude the proof. 
\end{proof}

\begin{remark}
From the proof of Lemma \ref{RIF}, we derive that $(-\Delta)^{s}_{p}u\in L^{\infty}(\mathbb{R}^{N})\cap C^{0}(\mathbb{R}^{N})$ for all $u\in C^{1, 1}(\mathbb{R}^{N})$, with $s<\frac{2(p-1)}{p}$ if $p\in (1, 2)$. 
\end{remark}


Next we prove a useful integration by parts formula for $W^{s, p}(\mathbb{R}^{N})\cap C^{1, 1}(\mathbb{R}^{N})$ functions and vector fields of class $C^{0, 1}_{c}(\mathbb{R}^N, \mathbb{R}^N)$.
\begin{lem}\label{LEMMA}
Let $N\in \mathbb{N}$, $s\in (0, 1)$ and $p\in (1, \infty)$. When $p\in (1, 2)$, we assume that $s<\frac{2(p-1)}{p}$. Let $u\in W^{s, p}(\mathbb{R}^{N})\cap C^{1, 1}(\mathbb{R}^{N})$ and $\mathcal{X}\in C^{0, 1}_{c}(\mathbb{R}^{N}, \mathbb{R}^{N})$.  
Then we have
\begin{align*}
&\frac{C_{N, s, p}}{2} \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \left[\dive(\mathcal{X}(x))+\dive(\mathcal{X}(y))-(N+sp) \frac{(\mathcal{X}(x)-\mathcal{X}(y))\cdot (x-y)}{|x-y|^{2}} \right]  dxdy\\
&=-p \int_{\mathbb{R}^{N}} \mathcal{X}(x)\cdot \nabla u(x) \, (-\Delta)^{s}_{p}u(x) \,dx.
\end{align*}
\end{lem}
\begin{proof}
We modify in a suitable way some arguments found in \cite[Lemma 4.2]{DFW1}. 
Exploiting the symmetry of the kernel $\frac{1}{|x|^{N+sp}}$ and Fubini's theorem, we see that 
\begin{align*}
&\frac{C_{N, s, p}}{2} \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \left[\dive(\mathcal{X}(x))+\dive(\mathcal{X}(y))-(N+sp) \frac{(\mathcal{X}(x)-\mathcal{X}(y))\cdot (x-y)}{|x-y|^{2}} \right]  dxdy \\
&\quad =-\frac{C_{N, s, p}(N+sp)}{2} \iint_{\mathbb{R}^{2N}} |u(x)-u(y)|^{p} \frac{(x-y)\cdot (\mathcal{X}(x)-\mathcal{X}(y))}{|x-y|^{N+sp+2}} dxdy\\
&\quad \quad+\frac{C_{N, s, p}}{2} \iint_{\mathbb{R}^{2N}}  |u(x)-u(y)|^{p} \frac{\dive \mathcal{X}(x)+\dive \mathcal{X}(y)}{|x-y|^{N+sp}}dxdy\\
&\quad =-\frac{C_{N, s, p}(N+sp)}{2} \lim_{\mu\rightarrow 0}\iint_{|x-y|>\mu} |u(x)-u(y)|^{p} \frac{(x-y)\cdot (\mathcal{X}(x)-\mathcal{X}(y))}{|x-y|^{N+sp+2}} dxdy\\
&\quad \quad+C_{N, s, p} \iint_{\mathbb{R}^{2N}}  |u(x)-u(y)|^{p} \frac{\dive \mathcal{X}(x)}{|x-y|^{N+sp}}dxdy\\
&\quad =-C_{N, s, p}(N+sp) \lim_{\mu\rightarrow 0} \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}\setminus \overline{B_{\mu}(y)}} |u(x)-u(y)|^{p}  \frac{(x-y)\cdot \mathcal{X}(x)}{|x-y|^{N+sp+2}}dxdy\\
&\quad \quad+C_{N, s, p} \iint_{\mathbb{R}^{2N}}  |u(x)-u(y)|^{p} \frac{\dive \mathcal{X}(x)}{|x-y|^{N+sp}}dxdy.
\end{align*}
Now, applying the divergence theorem in the domain $\mathbb{R}^{N}\setminus \overline{B_{\mu}(y)}$, with $y\in \mathbb{R}^{N}$ and $\mu>0$ fixed, and using the fact that 
$$
\nabla_{x}(|x-y|^{-(N+sp)})=-(N+sp)\frac{x-y}{|x-y|^{N+sp+2}},
$$ 
we have
\begin{align}\label{IBPF1}
&\frac{C_{N, s, p}}{2} \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \left[\dive(\mathcal{X}(x))+\dive(\mathcal{X}(y))-(N+sp) \frac{(\mathcal{X}(x)-\mathcal{X}(y))\cdot (x-y)}{|x-y|^{2}} \right]  dxdy \nonumber\\
&\quad=C_{N, s, p} \lim_{\mu\rightarrow 0} \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}\setminus \overline{B_{\mu}(y)}} |u(x)-u(y)|^{p} \, \nabla_{x}(|x-y|^{-(N+sp)}) \cdot \mathcal{X}(x) \, dxdy \nonumber\\
&\quad\quad+C_{N, s, p} \iint_{\mathbb{R}^{2N}}  |u(x)-u(y)|^{p} \frac{\dive \mathcal{X}(x)}{|x-y|^{N+sp}}dxdy \nonumber\\
&\quad=-C_{N, s, p} \lim_{\mu\rightarrow 0}\int_{\mathbb{R}^{N}} \int_{\mathbb{R}^{N}\setminus \overline{B_{\mu}(y)}} |u(x)-u(y)|^{p} \frac{\dive \mathcal{X}(x)}{|x-y|^{N+sp}}dxdy \nonumber\\
&\quad\quad -p C_{N, s, p} \lim_{\mu\rightarrow 0}\int_{\mathbb{R}^{N}} \int_{\mathbb{R}^{N}\setminus \overline{B_{\mu}(y)}} |u(x)-u(y)|^{p-2} (u(x)-u(y))  \frac{\nabla u(x)\cdot \mathcal{X}(x)}{|x-y|^{N+sp}}dxdy \nonumber\\
&\quad\quad +C_{N, s, p}\lim_{\mu\rightarrow 0}  \int_{\mathbb{R}^{N}} \int_{\partial B_{\mu}(y)} |u(x)-u(y)|^{p} \frac{(y-x)\cdot \mathcal{X}(x)}{|x-y|^{N+sp+1}} \, d\sigma(y) dx \nonumber\\
&\quad\quad+C_{N, s, p} \iint_{\mathbb{R}^{2N}}  |u(x)-u(y)|^{p} \frac{\dive \mathcal{X}(x)}{|x-y|^{N+sp}}dxdy \nonumber\\
&\quad=-pC_{N, s, p} \lim_{\mu\rightarrow 0} \iint_{|x-y|>\mu} |u(x)-u(y)|^{p-2}(u(x)-u(y)) \frac{\nabla u(x)\cdot \mathcal{X}(x)}{|x-y|^{N+sp}} dxdy\nonumber\\
&\quad\quad+C_{N, s, p} \lim_{\mu\rightarrow 0}  \mu^{-N-1-sp} \iint_{|x-y|=\mu} |u(x)-u(y)|^{p} (y-x)\cdot \mathcal{X}(x) d\sigma(x, y) \nonumber\\
&\quad=-\frac{pC_{N, s, p}}{2} \lim_{\mu\rightarrow 0} \int_{\mathbb{R}^{N}} \nabla u(x)\cdot \mathcal{X}(x) \int_{\mathbb{R}^{N}\setminus \overline{B_{\mu}(y)}} \left( \frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))}{|z|^{N+sp}} \right. \nonumber\\
&\quad\quad \left.+\frac{|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}}\right) dzdx\nonumber\\
&\quad\quad+\frac{C_{N, s, p}}{2} \lim_{\mu\rightarrow 0}  \mu^{-N-1-sp} \iint_{|x-y|=\mu} |u(x)-u(y)|^{p} (y-x)\cdot (\mathcal{X}(x)-\mathcal{X}(y)) d\sigma(x, y)\nonumber\\
&\quad=-p Y_{1}+Y_{2}.
\end{align} 
Since $u\in C^{1, 1}(\mathbb{R}^{N})$, it follows from Lemma \ref{RIF} that
\begin{align}\label{IBPF2}
Y_{1}&=\frac{C_{N, s, p}}{2}  \int_{\mathbb{R}^{N}}  \nabla u(x)\cdot \mathcal{X}(x) \int_{\mathbb{R}^{N}} \left(\frac{|u(x)-u(x+z)|^{p-2}(u(x)-u(x+z))}{|z|^{N+sp}}\right. \nonumber\\
&\quad \left. +\frac{|u(x)-u(x-z)|^{p-2}(u(x)-u(x-z))}{|z|^{N+sp}} \right) dzdx \nonumber\\
&=\int_{\mathbb{R}^{N}} \nabla u(x)\cdot \mathcal{X}(x)  \, (-\Delta)^{s}_{p}u(x) \,dx.
\end{align}
On the other hand, since $\mathcal{X}$ has compact support, we can find $R>0$ large enough such that $(\mathcal{X}(x)-\mathcal{X}(y))=0$ for all $x, y\in B_{R}(0)$ with $|x-y|<1$. For all $\mu\in (0, 1)$, we put
$$
\mathcal{N}_{\mu}=\Bigl\{(x, y)\in B_{R}(0)\times B_{R}(0): |x-y|=\mu \Bigr\}.
$$
Then, using $u\in C^{0, 1}(\mathbb{R}^{N})$, $\mathcal{X}\in C^{0, 1}(\mathbb{R}^{N}, \mathbb{R}^{N})$, and that the $2N-1$-dimensional measure of the set $\mathcal{N}_{\mu}$ is of order $O(N-1)$ as $\mu\rightarrow 0$, we have
\begin{align}\begin{split}\label{IBPF3}
&\mu^{-N-1-sp} \iint_{|x-y|=\mu} |u(x)-u(y)|^{p}(y-x)\cdot (\mathcal{X}(x)-\mathcal{X}(y)) d\sigma(x, y)\\
&\quad=\mu^{-N-1-sp} \iint_{\mathcal{N}_{\mu}} |u(x)-u(y)|^{p}(y-x)\cdot (\mathcal{X}(x)-\mathcal{X}(y)) d\sigma(x, y)=O(\mu^{p(1-s)})\rightarrow 0 \quad \mbox{ as } \mu\rightarrow 0,
\end{split}\end{align}
that is $Y_{2}=0$. Combining \eqref{IBPF1}, \eqref{IBPF2} and \eqref{IBPF3}, we obtain the desired formula.
\end{proof}
Now, we are ready to give the proof of Theorem \ref{thm2}.
\begin{proof}[Proof of Theorem \ref{thm2}]
Let $u\in W^{s, p}(\mathbb{R}^{N})\cap C^{1, 1}(\mathbb{R}^{N})$ be such that $(-\Delta)^{s}_{p}u=g(u)$ in $\mathbb{R}^{N}$. Pick $\varphi\in C^{1}_{c}(\mathbb{R}^{N})$ such that $0\leq \varphi\leq 1$ in $\mathbb{R}^{N}$, $\varphi(x)=1$ for $|x|\leq 1$ and $\varphi(x)=0$ for $|x|\geq 2$. Put $\varphi_{\lambda}(x)=\varphi(\lambda x)$ for all $x\in \mathbb{R}^{N}$ and $\lambda>0$.
Note that, for all $x\in \mathbb{R}^{N}$ and $\lambda>0$, 
\begin{align}\label{WS}
0\leq\varphi_{\lambda}(x)\leq 1 \,\mbox{ and } \, |x||\nabla \varphi_{\lambda}(x)|\leq C_{1},
\end{align}
 where the constant $C_{1}>0$ is independent of $\lambda$. 
Since
$\mathcal{X}(x)=\varphi_{\lambda}(x)x\in C^{0, 1}_{c}(\mathbb{R}^{N}, \mathbb{R}^{N})$ (note that \eqref{WS} implies that $|x\varphi_{\lambda}(x)-y\varphi_{\lambda}(y)|\leq C_{2}|x-y|$ for some constant $C_{2}>0$ independent of $\lambda$) and $u\in W^{s, p}(\mathbb{R}^{N})\cap C^{1, 1}(\mathbb{R}^{N})$, we can apply 
Lemma \ref{LEMMA} to deduce that 
\begin{align*}
&-\frac{C_{N, s, p}}{2p} \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \left[\dive(\mathcal{X}(x))+\dive(\mathcal{X}(y))-(N+sp) \frac{(\mathcal{X}(x)-\mathcal{X}(y))\cdot (x-y)}{|x-y|^{2}} \right]  dxdy\\
&=\int_{\mathbb{R}^{N}} \mathcal{X}(x)\cdot \nabla u(x) \, (-\Delta)^{s}_{p}u(x) \,dx\\
&=\int_{\mathbb{R}^{N}} \mathcal{X}(x)\cdot \nabla u(x) \, g(u(x)) \,dx.
\end{align*}
An integration by parts ensures that
\begin{align*}
-\int_{\mathbb{R}^{N}} g(u(x)) \varphi_{\lambda}(x)x\cdot \nabla u(x)dx=N \int_{\mathbb{R}^{N}} \varphi_{\lambda}(x)G(u(x))dx+\lambda \int_{\mathbb{R}^{N}} G(u(x)) x\cdot \nabla \varphi_{\lambda}(x)dx,
\end{align*}
and so
\begin{align}\label{BRUN1}
&-\frac{C_{N, s, p}}{2p} \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \left[\dive(\mathcal{X}(x))+\dive(\mathcal{X}(y))-(N+sp) \frac{(\mathcal{X}(x)-\mathcal{X}(y))\cdot (x-y)}{|x-y|^{2}} \right]  dxdy \nonumber\\
&=N \int_{\mathbb{R}^{N}} \varphi_{\lambda}(x)G(u(x))dx+\lambda \int_{\mathbb{R}^{N}} G(u(x)) x\cdot \nabla \varphi_{\lambda}(x)dx.
\end{align}
From \eqref{WS} we derive that
$$
\left| \dive(\mathcal{X}(x))+\dive(\mathcal{X}(y))-(N+sp) \frac{(\mathcal{X}(x)-\mathcal{X}(y))\cdot (x-y)}{|x-y|^{2}}\right|\leq C_{3},
$$
for some constant $C_{3}>0$ independent of $\lambda$. In view of this fact, $u\in W^{s, p}(\mathbb{R}^{N})$, \eqref{WS}, the pointwise convergences $\varphi_{\lambda}(x)\rightarrow 1$, $\mathcal{X}(x)\rightarrow x$ and $\dive\mathcal{X}(x)\rightarrow N$ for a.e. $x\in \mathbb{R}^N$, as $\lambda\rightarrow 0$, and $G(u)\in L^{1}(\mathbb{R}^{N})$, we can apply the dominated convergence theorem to infer that, as $\lambda\rightarrow 0$,
\begin{align}\label{BRUN2}
&-\frac{C_{N, s, p}}{2p} \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} \left[\dive(\mathcal{X}(x))+\dive(\mathcal{X}(y))-(N+sp) \frac{(\mathcal{X}(x)-\mathcal{X}(y))\cdot (x-y)}{|x-y|^{2}} \right]  dxdy \nonumber\\
&\rightarrow -\frac{C_{N, s, p}(N-sp)}{2p} \iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}} dxdy,
\end{align}
and
\begin{align}\label{BRUN3}
N \int_{\mathbb{R}^{N}} \varphi_{\lambda}(x)G(u(x))dx+\lambda \int_{\mathbb{R}^{N}} G(u(x)) x\cdot \nabla \varphi_{\lambda}(x)dx\rightarrow N \int_{\mathbb{R}^{N}} G(u(x))dx.
\end{align}
On account of \eqref{BRUN1}, \eqref{BRUN2} and \eqref{BRUN3}, we arrive at \eqref{POHOZAEV}.
\end{proof}







