\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 52, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/52\hfil Choquard type equations]
{Ground state solutions for Choquard type equations with a
singular potential}

\author[T. Wang \hfil EJDE-2017/52\hfilneg]
{Tao Wang}

\address{Tao Wang \newline
College of Mathematics and Computing Science,
Hunan University of Science and Technology,
Xiangtan, Hunan 411201, China. \newline
School of Mathematics and Statistics,
Central South University,
Changsha, Hunan 410083,  China}
\email{wt\_61003@163.com}

\dedicatory{Communicated by Claudianor O. Alves}

\thanks{Submitted June 6, 2016. Published February 21, 2017.}
\subjclass[2010]{35A15, 35A20, 35J20}
\keywords{Choquard equation; singular potential; ground state solution;
\hfill\break\indent Lions' concentration-compactness principle}

\begin{abstract}
 This article  concerns the  Choquard type equation
 $$
 -\Delta u+V(x)u=\Big(\int_{\mathbb{R}^N}\frac{|u(y)|^p}{|x-y|^{N-\alpha}}dy\Big)
 |u|^{p-2}u,\quad x\in \mathbb{R}^N,
 $$
 where $N\geq3$, $\alpha\in ((N-4)_+,N)$, $2\leq p<(N+\alpha)/(N-2)$
 and $V(x)$ is a possibly singular potential and  may be unbounded below.
 Applying a variant of the Lions' concentration-compactness principle,
 we prove the existence of ground state solution of the above equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In this article, we study  the  Choquard type equation
\begin{equation}\label{model-1}
-\Delta u+V(x)u=\Big(\int_{\mathbb{R}^N}\frac{|u(y)|^p}{|x-y|^{N-\alpha}}dy\Big)
 |u|^{p-2}u,\quad x\in \mathbb{R}^N,
\end{equation}
where $N\geq3$, $\alpha\in((N-4)_+,N)$, $p\in[2,\frac{N+\alpha}{N-2})$ and
$V$ is a given potential satisfying the following assumptions
\begin{itemize}
\item[(A1)] $V:\mathbb{R}^N\to \mathbb{R}$ is a measurable function;

\item[(A2)] $V_\infty :=\lim_{|y|\to\infty}V(y)\geq V(x)$,
 for almost every $x\in \mathbb{R}^N$, and the inequality is
 strict in a non-zero measure domain;

\item[(A3)] there exists $\bar{C}>0$ such that for any $u\in H^1(\mathbb{R}^N)$,
$$
\int_{\mathbb{R}^N}(|\nabla u|^2+V(x)|u|^2)dx
\geq \bar{C}\Big(\int_{\mathbb{R}^N}(|\nabla u|^2+|u|^2)dx\Big).
$$
\end{itemize}
Clearly, (A3) implies $V_\infty>0$. When $N=3$, $\alpha=2$, $p=2$,
\eqref{model-1} with  $V\equiv 1$ just is the classical stationary
 Choquard equation
 \begin{equation}\label{model-10}
-\Delta u+u=\Big(\int_{\mathbb{R}^N}\frac{|u(y)|^2}{|x-y|}dy\Big)u\quad
 \text{in }  \mathbb{R}^3.
\end{equation}
This equation appeared at least as early as in 1954, in a work by
Pekar describing the quantum mechanics of a polaron at rest \cite{sp}.
In 1976, Choquard used \eqref{model-10} to describe an electron trapped in its
own hole, in a certain approximation to Hartree-Fock theory of one component
plasma \cite{LEHS}. As is known to us, the existence and multiplicity of radial
solutions to \eqref{model-10} has been studied in \cite{LEH} and \cite{LPL}.
Further more results for related problems  can be founded in
\cite{an, cms, csv, pl, ldf, mgo, nm, tm, ww} and references therein,
where $V$ may be not  a positive constant.

In recent years,  the existence and  properties of solutions for the generalized
Choquard type equation \eqref{model-1} are  widely considered.
 When the potential $V$ is a positive constant, Ma and Zhao  \cite{MZ}  proved
the positive solutions for the generalized Choquard equation \eqref{model-1}
must be radially symmetric and monotone decreasing about some point under
appropriate assumptions on $p,\alpha, N$. They also showed the positive solutions
 of \eqref{model-10} is uniquely determined, up to translations, see also \cite{ccs}.
Moroz and Van Schaftingen \cite{vv} obtained the existence,  regularity,
positivity and radial symmetry of ground state solution  of  \eqref{model-1}, and
they also derived  the sharp decay asymptotic of the ground state solution.
 For more related problems, one can  see \cite{gmvs,LY, lgb, vj}. When  the
potential  $V$ is continuous and bounded below in $\mathbb{R}^N$,
Alves and Yang \cite{acy}  studied the multiplicity and concentration
behaviour of positive solutions for quasilinear Choquard equation
\begin{equation}\label{minmax36}
-\epsilon^p\Delta_p u+V(x)|u|^{p-2}u
=\epsilon^{\mu-N}\Big(\int_{\mathbb{R}^N}\frac{Q(y)F(u(y))}{|x-y|^{\mu}}dy\Big)
Q(x)f(u)\ \text{in}\ \mathbb{R}^N,
\end{equation}
where $\Delta_p$ is the $p$-Laplacian operator, $1<p<N$, $V$ and $Q$ are two
continuous real functions on $\mathbb{R}^N$, $F(s)$ is the primitive
function of $f(s)$ and $\epsilon$ is a positive parameter. Furthermore results
for related problems can be found in \cite{acoy,acoyf, css, vjv,yhy} and
references therein.

To the best of our knowledge, there are only a few results
on the existence of ground state solutions of \eqref{model-1} with singular
potentials which may be unbounded below. In this paper, we succeed in
finding a ground state solution of \eqref{model-1} under the assumptions
(A1)--(A3). Here we remark the assumptions are introduced in \cite{aa}
to study the singular nonlinear Schr\"{o}dinger-Maxwell equations.
Our aim is to extend the results in \cite{aa} to the case of  Choquard
type equations with some new techniques.
Recall that  $u\in H^1(\mathbb{R}^N)$ is said to be a ground state solution to
\eqref{model-1}, if $u$ solves  \eqref{model-1} and  minimizes the energy
functional associated with \eqref{model-1} among all possible nontrivial solutions.

\begin{remark} \label{rmk1.1} \rm
 According to \cite[Remark 1.3]{aa}, we can conclude that
the  potential $V$ can be satisfied by the following type of functions
which are singular and unbounded below.
Let $V(x)=\gamma(x)-\lambda|x|^{-\sigma}$. Here $\gamma$ satisfies
(A1) and (A2), and is bounded below by a positive constant,
$\sigma\in (0,2]$ and $\lambda$
is a positive constant small enough.
Indeed,  we only need to verify (A3). By Hardy's inequality (see \cite{els}),
there exists $C>0$ such that
$$
\int_{\mathbb{R}^N}\frac{u^2}{|x|^2}dx\leq C\int_{\mathbb{R}^N}|\nabla u|^2dx.
$$
Throughout this article,  we write  $C$ for different positive constants.
If $\sigma\in (0,2)$, using H\"{o}lder's inequality, we have
\begin{equation}
\begin{aligned}
\int_{\mathbb{R}^N}\frac{u^2}{|x|^\sigma}dx
&\leq \Big(\int_{\mathbb{R}^N}|\frac{u^\sigma}{|x|^\sigma}|^{\frac{2}{\sigma}}
 dx\Big)^{\sigma/2}
\Big(\int_{\mathbb{R}^N}|u^{2-\sigma}|^{\frac{2}{2-\sigma}}
 \Big)^{\frac{2-\sigma}{2}}\\
&\leq C \Big(\int_{\mathbb{R}^N}|\nabla u|^2dx\Big)^{\sigma/2}
\Big(\int_{\mathbb{R}^N}|u|^2dx\Big)^{\frac{2-\sigma}{2}}\\
&\leq C\Big(\int_{\mathbb{R}^N}(|\nabla u|^2+|u|^2)dx\Big).
\end{aligned}
\end{equation}
By taking $\lambda>0$ small enough,  (A3) holds immediately.
In particular, $\gamma(x)\equiv$ positive constant.
\end{remark}

Now  we are ready to state our main results.

\begin{theorem}\label{thm1.1}
Let $N\geq3,\alpha\in((N-4)_+,N),p\in [2,\frac{N+\alpha}{N-2})$ and suppose
{\rm  (A1)--(A3)} hold.
Then  \eqref{model-1} has  a ground state solution in $H^1(\mathbb{R}^N)$.
\end{theorem}

The remainder of this article is organized as follows.
In Section 2, some preliminary results are presented.
In  Section 3, we are devoted to the proof of our main result.

\section{Preliminary results} \label{Sec2}

In this article, we  use the following  notation.\\
$\bullet$ Let $N$  be positive integers and $B_R$
 be an open ball of radius $R$  centered at the origin in $\mathbb{R}^N$.\\
$\bullet$ Let $H^1(\mathbb{R}^N)$ be the usual Sobolev space with the standard norm
$$
\|u\|_H=\Big(\int_{\mathbb{R}^N}(|\nabla u|^2+|u|^2)dx\Big)^{1/2}.
$$
We also use the notation
$$
\|u\|=\Big(\int_{\mathbb{R}^N}(|\nabla u|^2+V(x)|u|^2)dx\Big)^{1/2},
$$
which is a norm equivalent to $\|\cdot\|_{H}$ in $H^1(\mathbb{R}^N)$ under
(A1)--(A3) (we will prove the equivalence in Lemma~\ref{norm3}).\\
$\bullet$ Let $\Omega\subset \mathbb{R}^N$ be a domain.  For $1\leq s<\infty$,
$L^s(\Omega)$ denotes the Lebesgue space with the norm
  $$
|u|_{L^s(\Omega)}=\Big(\int_{\Omega} |u|^sdx \Big)^{1/s}.
$$
 If $\Omega=\mathbb{R}^N$, we write $|u|_{L^s}=|u|_{L^s(\Omega)}$.
We can identify $u\in L^s(\Omega)$ with its extension to $\mathbb{R}^N$
obtained by setting $u=0$ in $\mathbb{R}^N\backslash\Omega$, which ensures
that we can use Hardy-Littlewood-Sobolev inequality to deal with the nonlocal
problem.\\
$\bullet$ The dual space of $H^1(\mathbb{R}^N)$ is denoted by $H^{-1}(\mathbb{R}^N)$.
 The norm on $H^{-1}(\mathbb{R}^N)$ is denoted by
$\|\cdot\|_{H^{-1}}$.

It is well known that the energy functional
$I: H^1(\mathbb{R}^N)\to \mathbb{R}$ associated with \eqref{model-1} is defined  by
 $$
I(u)=\frac{1}{2}\|u\|^2-
 \frac{1}{2p}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy.
$$
This is a well defined $C^2(H^1(\mathbb{R}^N),\mathbb{R})$ functional whose
 Gateaux derivative is given by
$$
I'(u) v=\int_{\mathbb{R}^N}(\nabla u \nabla v+V(x)uv)dx
-\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(y)|^p
|u(x)|^{p-2}u(x)v(x)}{|x-y|^{N-\alpha}}\,dx\,dy
$$
for all $v\in H^1(\mathbb{R}^N)$.
It is easy to see that all solutions of \eqref{model-1} correspond to
critical points of the energy functional  $I$.
 For simplicity of notation, we  write
$$
\mathbb{D}(u)=\int_{\mathbb{R}^N}
\int_{\mathbb{R}^N}\frac{|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy.
$$
To study the  nonlocal problems related with \eqref{model-1}, we need to
recall the following well-known Hardy-Littlewood-Sobolev inequality.

\begin{lemma}[see \cite{llm}] \label{lem2.3}
Let $s,t>1$ and $0<\mu<N$ with $\frac{\mu}{N}+\frac{1}{s}+\frac{1}{t}=2$,
$f\in L^s(\mathbb{R}^N)$ and $h\in L^t(\mathbb{R}^N)$. There exists
a sharp constant $C(N,\mu,s,t)$ independent of $f,h$, such that
$$
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{f(x)h(y)}{|x-y|^\mu}\,dx\,dy
\leq C(N,\mu,s,t)|f|_{L^s}|h|_{L^t}.
$$
\end{lemma}

\begin{remark} \label{rmk2.1} \rm
Suppose
$\frac{N+\alpha}{N}\leq p\leq\frac{N+\alpha}{N-2}$ and
$u\in L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$.
Then according to Lemma~\ref{lem2.3}, we have
\begin{equation}\label{nonlocal}
\mathbb{D}(u)\leq C(N,\alpha,s,t,)|u^{p}|_{L^s}|u^{p}|_{L^t}
=C(N,\alpha,s,t)|u|^{p}_{L^{\frac{2Np}{N+\alpha}}}
|u|^{p}_{L^{\frac{2Np}{N+\alpha}}}<\infty.
\end{equation}
where $C$ only depends on $N,s,t,\alpha$ and
$\frac{1}{s}+\frac{1}{t}+\frac{N-\alpha}{N}=2$.
\end{remark}

Using \cite[Theorem 1 and Proposition 5]{vv}, we easily obtain the following lemma.

\begin{lemma}\label{groundstate1}
Let $N\geq3$, $\alpha\in(0,N)$,  $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2})$.
Then there exists a positive ground state solution $w\in H^1(\mathbb{R}^N)$
of \eqref{model-1} with $V\equiv$ positive constant.
\end{lemma}

According to the assumptions (A1)--(A3), we have the following lemma
whose proof is standard.

\begin{lemma}\label{norm3}
For any $u\in H^1(\mathbb{R}^N)$, there exist two positive constants
$C_1$ and $C_2$ such that
\begin{equation}\label{norm1}
C_1\|u\|_{H}\leq\|u\|\leq  C_2\|u\|_{H}.
\end{equation}
\end{lemma}

In this article, we define the Nehari manifold
$$
\mathcal{N}=\{u\in H^1(\mathbb{R}^N)\backslash\{0\}:I'(u)u=0\}.
$$
Let
$$
c:=\inf_{u\in\mathcal{N}} I(u).
$$
 It is easy to check that $0\notin \partial\mathcal{N}$
and $c>0$.
Now we show  some  properties of the Nehari manifold $\mathcal{N}$.

\begin{lemma}\label{lem2.4}
Suppose {\rm (A1)--(A3)}  hold. Then the following statements hold:
\item(i) For every $u\in H^1(\mathbb{R}^N)\backslash\{0\}$,
there exists a unique $t_u\in (0,\infty)$ such that $t_uu\in \mathcal{N} $ and
$t_u=\left(\frac{\|u\|^2}{\mathbb{D}(u)}\right)^\frac{1}{2p-2}$.
Furthermore,
$$
I(t_u u)=\sup_{t>0}I(tu)=(\frac{1}{2}-\frac{1}{2p})
\Big(\frac{\|u\|^2}{\mathbb{D}^{\frac{1}{p}}(u)}\Big)^{\frac{p}{p-1}}.
$$

\item(ii) $c=\inf_{u\in\mathcal{N}} I(u)
=\inf_{u\in H^1(\mathbb{R}^N)\backslash\{0\}}\sup_{t>0}I(tu)$.
\end{lemma}

\begin{proof}
Statement (i) follows by a direct calculation.
Then by (i), we have
$I(t_u u)=\sup_{t>0}I(tu)\geq\inf_{u\in \mathcal{N}}I(u)$. Hence
$\inf_{u\in H^1(\mathbb{R}^N)\backslash \{0\}}\sup_{t>0} I(tu)
\geq\inf_{u\in \mathcal{N}} I(u)$.
On the other hand,
for any  $u\in\mathcal{N}$,
$$
I(u)=\sup_{t>0}I(tu)\geq\inf_{u\in H^1(\mathbb{R}^N)\backslash\{0\}}\sup_{t>0}I(tu).
$$
This shows (ii) and completes the proof.
\end{proof}

Let $\lambda>0$. We define
\begin{gather}
I_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^N}(|\nabla u|^2+\lambda u^2)dx-
 \frac{1}{2p}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)|^p
|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy, \nonumber\\
\label{model-001}
c(\lambda)=\inf_{u\in N_\lambda}I_\lambda(u),
\end{gather}
where $\mathcal{N}_\lambda$ is the Nehari manifold of $I_{\lambda}$.
Then we give some preliminary lemmas which can be proved by using the
 similar arguments as in \cite[Lemma 2.7]{aap} with
some necessary modifications.

\begin{lemma}\label{lem2.5}
Let $c(\lambda)$ be defined in \eqref{model-001}. Then $c(\lambda)$ is a
continuous and strictly increasing function in $(0,\infty)$.
\end{lemma}

\begin{proof}
Let $\lambda,\delta,\lambda_n>0$.
We first show $c(\lambda)$ is strictly increasing with respect to $\lambda$.
To be precise, if $\lambda<\delta$, we have $c(\lambda)<c(\delta)$.

Indeed, according to Lemma~\ref{groundstate1}, there exists
$u\in H^1(\mathbb{R}^N)$ such that
$u$ is a positive critical point of $I_\delta$ and $I_\delta(u)=c(\delta)$.
On the other hand,  by Lemma \ref{lem2.4} (i),
we can find  a unique $t_u>0$ such that $t_uu\in \mathcal{N}_\lambda$. Then
\begin{equation}\label{model-2}
\begin{aligned}
c(\delta)&=I_\delta(u)\geq I_\delta(t_uu)\\
&=I_\lambda(t_uu)+(\delta-\lambda)\int_{\mathbb{R}^N}|t_uu|^2dx\\
&\geq c(\lambda)+(\delta-\lambda)\int_{\mathbb{R}^N}|t_uu|^2dx.
\end{aligned}
\end{equation}
So if $\lambda<\delta$, it holds $c(\lambda)<c(\delta)$.

Now we prove $c(\lambda)$ is continuous with respect to $\lambda$. To be precise,
if $\lambda_n\to \lambda$, then $c(\lambda_n)\to c(\lambda)$.

Let $\lambda_n=\lambda+h_n$, where $h_n\to 0 $ as $n\to\infty$.
It suffices to prove $c(\lambda+h_n)\to c(\lambda)$ as $n\to \infty$.
We shall complete the proof by distinguishing two cases.
\smallskip

\noindent\textbf{Case 1.}  We show that
$$
c^+:=\lim_{h_n\to 0^+}c(\lambda+h_n)=c(\lambda).
$$
In fact, according to the monotonicity  of $c(\lambda)$,  we have
$c^+\geq c(\lambda)>0$. By way of contradiction,  suppose
\begin{equation}\label{model-3}
c^+>c(\lambda).
\end{equation}
By Lemma \ref{groundstate1},  there exists a positive function
 $u\in H^1(\mathbb{R}^N)$ such that
$I_\lambda'(u)=0$ and  $I_\lambda(u)=c(\lambda)$. In addition, for each $n$,
there exists a unique $\theta_n>0$ such that $\theta_nu\in \mathcal{N}_{\lambda_n}$
 due to Lemma \ref{lem2.4} (i).
Note that
\begin{gather*}
\int_{\mathbb{R}^N}|\nabla u|^2+\lambda|u|^2=\mathbb{D}(u)\\
\int_{\mathbb{R}^N}|\nabla u|^2+\lambda_n|u|^2=\theta_n^{2p-2}\mathbb{D}(u).
\end{gather*}
Then a standard argument shows that $(\theta_n)_{n\geq1}$ is uniformly bounded.
In addition, by Lemma~\ref{lem2.4} (i) and Sobolev embedding theorem,
we have
\begin{align*}
c^+
&\leq c(\lambda+h_n)\leq I_{\lambda_n}(\theta_nu)\\
&= I_\lambda(\theta_nu)+\frac{1}{2}(\lambda_n-\lambda)
 \int_{\mathbb{R}^N}|\theta_nu|^2\\
&\leq I_\lambda(u)+\frac{1}{2}(\lambda_n-\lambda)\int_{\mathbb{R}^N}|\theta_nu|^2\\
&= c(\lambda)+\frac{1}{2}(\lambda_n-\lambda)\int_{\mathbb{R}^N}|\theta_nu|^2\\
&\leq c(\lambda)+Ch_n\|\theta_nu\|_H^2.
\end{align*}
Letting $n\to \infty$, we conclude $c^+\leq c(\lambda)$, a contradiction
to \eqref{model-3}.
\smallskip

\noindent\textbf{Case 2.} We shall prove
$$
c^-:=\lim_{h_n\to 0^{-}}c(\lambda+h_n)=c(\lambda).
$$
Indeed, the monotonicity  of $c(\lambda)$ yields $c^- \leq c(\lambda)$.
By way of contradiction, we suppose
\begin{equation}\label{model-4}
c^{-}<c(\lambda).
\end{equation}
By Lemma \ref{groundstate1},  for each $n\geq1$, there exists a positive
function $v_n\in H^1(\mathbb{R}^N)$ such that $I'_{\lambda_n}(v_n)=0$ and
$I_{\lambda_n}(v_n)=c(\lambda_n)$.
Since $c(\frac{\lambda}{2})\leq c(\lambda_n)=I_{\lambda_n}(v_n)\leq c(\lambda)$
for $n$ large enough, we can find $C_1,C_2>0$ such that
$C_1\leq\|v_n\|_H\leq C_2$ uniformly  in $H^1(\mathbb{R}^N)$.
By Lemma \ref{lem2.4} (i), for each $n\geq1$, there exists a unique
$\theta(v_n)>0$ such that $\theta(v_n)v_n\in \mathcal{N}_{\lambda}$.
Note that
\begin{gather*}
\int_{\mathbb{R}^N}|\nabla v_n|^2+\lambda_n|v_n|^2=\mathbb{D}(v_n)\\
\int_{\mathbb{R}^N}|\nabla v_n|^2+\lambda|v_n|^2=\theta^{2p-2}(v_n)\mathbb{D}(v_n).
\end{gather*}
Then a standard argument shows that $(\theta(v_n))_{n\geq1}$ is uniformly  bounded.
In addition, by Lemma~\ref{lem2.4} (i) and Sobolev embedding theorem,
we have
\begin{align*}
c(\lambda)&\leq  I_{\lambda}(\theta(v_n)v_n)\\
&= I_{\lambda_n}(\theta(v_n)v_n)+\frac{1}{2}(\lambda-\lambda_n)\int_{\mathbb{R}^N}|\theta(v_n)v_n|^2\\
&\leq I_{\lambda_n}(v_n)+\frac{1}{2}(\lambda-\lambda_n)\int_{\mathbb{R}^N}|\theta(v_n)v_n|^2\\
&= c(\lambda_n)+\frac{1}{2}(\lambda-\lambda_n)\int_{\mathbb{R}^N}|\theta(v_n)v_n|^2\\
&\leq c(\lambda_n)+Ch_n\|\theta(v_n)v_n\|_H^2.
\end{align*}
Since $\lim_{n\to\infty}c(\lambda_n)=c^{-}$, letting $n\to \infty$,
we conclude $ c(\lambda)\leq c^-$, a contradiction to \eqref{model-4}.
The proof is complete.
\end{proof}

\begin{lemma}\label{lem2.6}
 Let {\rm (A1)--(A3)}  hold. Then $c<c(V_\infty)$. Moreover, there exists
 $\mu>0$ such that $c<c(V_\infty-\mu)<c(V_\infty)$.
\end{lemma}

\begin{proof}
By Lemma~\ref{groundstate1},  there exists a positive function 
$u\in H^1(\mathbb{R}^N)$ such that
$I_{V_\infty}'(u)=0$ and  $I_{V_\infty}(u)=c(V_\infty)$. 
In addition, there exists a unique $t_u>0$ such that
$t_uu\in \mathcal{N}$. By (A2), we obtain
\begin{equation}\label{model-002}
\begin{aligned}
c(V_\infty)
&= I_{V_\infty}(u)\geq I_{V_\infty}(t_uu)\\
&= I(t_uu)+\int_{\mathbb{R}^N}(V_\infty-V(x))|t_uu|^2dx\\
&\geq c+\int_{\mathbb{R}^N}(V_\infty-V(x))|t_uu|^2dx >c.
\end{aligned}
\end{equation}
By Lemma~\ref{lem2.5},  there exists $\mu>0$ such that
$$
|c(V_\infty)-c(V_\infty-\mu)|<\frac{c(V_\infty)-c}{2},
$$
which  implies 
$c(V_\infty-\mu)>c$. In addition, we have $c(V_\infty-\mu)<c(V_\infty)$.
This completes the proof.
\end{proof}

\section{Proof of Theorem \ref{thm1.1}}
In this section, motivated by \cite{aa} and \cite{csl}, we shall  prove 
the existence of ground state solution of \eqref{model-1} by using 
a variant of Lions' concentration-compactness principle.

Let $(u_n)_{n\geq1}\subset\mathcal{N}$ be a minimizing sequence such that
\begin{equation}\label{minimizing}
\lim_{n\to\infty}I(u_n)=c.
\end{equation} 
In what follows, we shall prove
$(u_n)_{n\geq1}$ is a $(PS)_c$ sequence of $I$.

\begin{definition} \label{def3.1} \rm
 We say that $(u_n)_{n\geq1}\subset H^1(\mathbb{R}^N)$ is $(PS)_c$ sequence of $I$, 
if $(u_n)_{n\geq1}$ satisfies
\begin{equation}\label{model-20}
I(u_n)\to c, I'(u_n)\to 0,\quad\text{as }n\to \infty.
\end{equation}
\end{definition}

\begin{lemma}\label{lem3.1}
If $(u_n)_{n\geq1}\subset \mathcal{N}$ is a minimizing sequence such that 
\eqref{minimizing} holds, then
 $(u_n)_{n\geq1}$ is a $(PS)_c$ sequence of $I$.
\end{lemma}

\begin{proof}
The outline of the proof is as follows.
\smallskip

\noindent\textbf{Step 1.} We shall show $G'(u_n)\neq 0$ for any $n\geq1$.
In fact,  it is easy to check $(u_n)_{n\geq1}$ is uniformly bounded in 
$H^1(\mathbb{R}^N)$.
Let $G: H^1(\mathbb{R}^N)\to \mathbb{R}$ defined by
$$ 
G(u)=\|u\|^2-\mathbb{D}(u).
$$
By standard arguments, we deduce $G$ is of class 
$\mathcal{C}^2(H^1(\mathbb{R}^N),\mathbb{R})$ and its Gateaux derivative is 
given by
$$
G'(u)v=2 \int_{\mathbb{R}^N}(\nabla u \nabla v+V(x)uv)\,dx\,dy
-2p\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(y)|^p|u(x)|^{p-2}
u(x)v(x)}{|x-y|^{N-\alpha}}\,dx\,dy
$$
for all $u,v$ in $H^1(\mathbb{R}^N)$.  Since $(u_n)_{n\geq1}$ is  uniformly 
bounded, by \eqref{nonlocal}, we deduce $G'(u_n)$ is uniformly bounded in 
$H^{-1}(\mathbb{R}^N)$. Note that for any $u\in \mathcal{N}$,
$$
I(u)=(\frac{1}{2}-\frac{1}{2p})\|u\|^2\geq c.
$$
Then, for all $u\in \mathcal{N}$,
$$
G'(u)u=2\|u\|^2-2p\mathbb{D}(u)=(2-2p)\|u\|^2\leq -4pc<0.
$$ 
Hence $G'(u_n)\neq 0$.
\smallskip

\noindent\textbf{Step 2.} 
We shall prove $I'(u_n)\to 0$ as $n\to\infty$. 
Let $J_{\lambda}(u_n):=\|I'(u_n)-\lambda G'(u_n)\|_{H^{-1}}$. 
By \cite[Theorem 8.5]{wm},  we assume
$$
\min_{\lambda\in \mathbb{R}}J_{\lambda}(u_n)\to 0,\quad \text{as } n\to\infty.
$$
Then up to a subsequence, we have
$$
\min_{\lambda\in \mathbb{R}}J_{\lambda}(u_n)<\frac{1}{2n}.
$$
On the other hand, for each $n$, we can find $\lambda_n\in \mathbb{R}$ such that
$$
|J_{\lambda_n}(u_n)-\min_{\lambda\in \mathbb{R}}J_{\lambda}(u_n)|<\frac{1}{2n}\,.
$$
Therefore, 
\begin{equation}\label{ps}
J_{\lambda_n}(u_n)=\|I'(u_n)-\lambda_n G'(u_n)\|_{H^{-1}}\to 0,\quad
\text{as } n\to\infty.
\end{equation}
Note that
\begin{equation}
|I'(u_n)u_n-\lambda_n G'(u_n)u_n|
\leq \|I'(u_n)-\lambda_n G'(u_n)\|_{H^{-1}}\|u_n\|.
\end{equation}
Since $I'(u_n)u_n=0$ and $G'(u_n)u_n\neq 0$, we conclude from \eqref{ps} that 
$\lambda_n\to 0$ as $n\to\infty$. Note that $G'(u_n)\neq0$ by Step 1. Then we have
\[
\|I'(u_n)\|_{H^{-1}}\leq\|I'(u_n)-\lambda_n G'(u_n)\|_{H^{-1}}
+\|\lambda_n G'(u_n)\|_{H^{-1}}\to0
\]
as $n\to\infty$. This completes the proof.
\end{proof}

Next we need  some  compactness on the minimizing  sequence
 $(u_n)_{n\geq1}$ defined in \eqref{minimizing} in order to prove our
 existence results.
Define the functional $J:H^1(\mathbb{R}^N)\to\mathbb{R}$ by
$$
J(u)=(\frac{1}{2}-\frac{1}{2p})\int_{\mathbb{R}^N}|\nabla u|^2+V(x)|u|^2.
$$
It is easy to check that for any $u\in\mathcal{N}$,
we have $I(u)=J(u)$. Using a variant of  Lion's concentration-compactness
 principle  presented in \cite[Lemma 6.1]{csl}
(see also \cite[Proposition 3.1]{lm}), we have the following
lemma. Throughout this section,  $O(\mu)$ denotes a constant depending on
 $\mu$ such that $|\frac{O(\mu)}{\mu}|\leq C$.

\begin{lemma}\label{lem3.2}
For any $\epsilon>0$, there exists $\bar{R}=\bar{R}(\epsilon)>0$ such that for 
any $n\geq \bar{R}$,
$$
\int_{|x|>\bar{R}}(|\nabla u_n|^2+|u_n|^2)<\epsilon.
$$
\end{lemma}

\begin{proof}
By way of contradiction, we suppose that there exist $\epsilon_0>0$ and a 
subsequence $(u_k)_{k\geq 1}$
such that for any $k\geq1$,
\begin{equation}\label{compact1}
\int_{|x|>k} (|\nabla u_k|^2+|u_k|^2)\geq \epsilon_0.
\end{equation}
Let
$$
\rho_k(\Omega)=\int_{\Omega}(|\nabla u_k|^2+|u_k|^2).
$$
Fix  $l>1$ and define
$$
A_r:=\{x\in \mathbb{R}^N|r\leq|x|\leq r+l\}, \quad \text{for any }\ r>0.
$$
We shall finish the proof by distinguishing four steps.
\smallskip

\noindent\textbf{Step 1.} We shall show that
 for any $\mu,R>0$, there exists $r=r(\mu,R)>R$ such that $\rho_k(A_r)<\mu$
for infinitely many $k$.

We argue by contradiction. Suppose there exist $\mu_0>0$ and 
$\tilde{R}\in \mathbb{N}$
such that, for any $m\geq\tilde{R}$, there exists a strictly increasing sequence 
$\{p(m)\}_{m\geq\tilde{R}}\subset (0,\infty)$ such that
$$
\rho_k(A_m)\geq \mu_0,\quad \text{for any } k\geq p(m).
$$
By applying this fact, we have
$$
\|u_k\|^2_H\geq \rho_k(B_m\backslash B_{\tilde{R}})
\geq \big(\frac{m-\tilde{R}}{l}\big)\mu_0,\quad \text{for any }
 m\geq \tilde{R},\;  k\geq p(m).
$$
Take $m>\tilde{R}+\frac{l}{\mu_0}\sup_{n\geq1}\|u_n\|_{H}^2$. Then
$$
\|u_{p(m)}\|_H^2>\sup_{n\geq1}\|u_n\|_{H}^2\geq\|u_{p(m)}\|_H^2,
$$
which is a contradiction.
\smallskip

\noindent\textbf{Step 2.}
 We shall  show there exists $\mu_0\in(0,1)$ such that for any $\mu\in (0,\mu_0)$,
we have the following results:
\item(i) It holds
\begin{equation}\label{compact3}
c<c(V_\infty-\mu)<c(V_\infty).
\end{equation}
\item(ii) There exists $R_\mu>0$ such that for almost every $|x|>R_\mu$,
\begin{equation}\label{compact4}
V(x)\geq V_\infty-\mu>0.
\end{equation}
\item(iii) There exists $r>R_\mu$ such that, going if necessary to a subsequence,
\begin{equation}\label{compact5}
\rho_k(A_r)<\mu,\quad \text{for all } k\geq1.
\end{equation}
\item(iv) It holds
\begin{gather}\label{compact6}
\int_{A_r}|\nabla u_k|^2+V(x)|u_k|^2=O(\mu),\quad \text{for all } k\geq 1, \\
\label{compact7}
\int_{A_r}\int_{\mathbb{R}^N}\frac{|u_k(x)|^p|u_k(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy
=O(\mu),\quad  \text{for all } k\geq 1.
\end{gather}

Indeed, (i) follows from  Lemma~\ref{lem2.6}.  By $(V2)$, we can find 
$R_\mu>0$ such that for almost every $|x|>R_\mu$, \eqref{compact4} holds.
 Consider $\mu$  and $R_{\mu}$ satisfying \eqref{compact3} and \eqref{compact4}. 
Then by Step 1, we can take $r>R_\mu$ such that, going if necessary to a subsequence, 
\eqref{compact5} is valid. According to  (A2), \eqref{compact4}and \eqref{compact5},
 we can easily obtain \eqref{compact6}. Since $\mu\in(0,1)$, combining 
\eqref{nonlocal}, we  have \eqref{compact7}.
\smallskip

\noindent\textbf{Step 3.} 
We shall first give some estimates. Let $\eta\in C^\infty(\mathbb{R}^N)$ such 
that $\eta=1 $ in $B_r$ and $\eta=0$ in ${B^c_{r+l}}$, $0\leq \eta\leq1$
and $|\nabla \eta|\leq 2$, where $r$ is defined in Step 2 (iii).
Define $v_k=\eta u_k$ and $w_k=(1-\eta) u_k$.

It follows from \eqref{compact4} and \eqref{compact6} that
\begin{equation}\label{compact18}
\begin{gathered}
\int_{A_r}|\nabla v_k|^2+V(x)|v_k|^2=O(\mu),\\
\int_{A_r}|\nabla w_k|^2+V(x)|w_k|^2=O(\mu).
\end{gathered}
\end{equation}
This, combined with \eqref{compact6}, implies that
\begin{equation}\label{compact9}
\begin{aligned}
&\int_{\mathbb{R}^N}|\nabla u_k|^2+V(x)|u_k|^2 \\
&= \int_{A_r}|\nabla u_k|^2+V(x)|u_k|^2+\int_{B_r}|\nabla v_k|^2+V(x)|v_k|^2
 +\int_{B^c_{r+l}}|\nabla w_k|^2+V(x)|w_k|^2\\
&= \int_{\mathbb{R}^N}|\nabla v_k|^2+V(x)|v_k|^2
 +\int_{\mathbb{R}^N}|\nabla w_k|^2+V(x)|w_k|^2
 +\int_{A_r}|\nabla u_k|^2+V(x)|u_k|^2\\
&\quad -\int_{A_r}|\nabla v_k|^2+V(x)|v_k|^2
 -\int_{A_r}|\nabla w_k|^2+V(x)|w_k|^2\\
&= \int_{\mathbb{R}^N}|\nabla v_k|^2+V(x)|v_k|^2
 +\int_{\mathbb{R}^N}|\nabla w_k|^2+V(x)|w_k|^2+O(\mu).
\end{aligned}
\end{equation}
According to Step 1 above, we can take $l>0$ appropriately large such that
$$
\int_{B_r}\int_{B^c_{r+l}}\frac{|u_k(x)|^p|u_k(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy
=O(\mu).
$$
Then we conclude from \eqref{compact6}, \eqref{compact7} and \eqref{compact18} that
\begin{equation}\label{compact13}
\begin{aligned}
&\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u_k(x)|^p
 |u_k(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy \\
&= \int_{B_r}\int_{B_r}\frac{|u_k(x)|^p|u_k(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy
+2\int_{B_r}\int_{B^c_{r+l}}\frac{|u_k(x)|^p|u_k(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy\\
&\quad +\int_{B_r}\int_{A_r}\frac{|u_k(x)|^p|u_k(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy
 +\int_{B^c_{r+l}}\int_{B^c_{r+l}}\frac{|u_k(x)|^p|u_k(y)|^p}{|x-y|^{N-\alpha}}
 \,dx\,dy\\
&\quad +\int_{B^c_{r+l}}\int_{A_r}\frac{|u_k(x)|^p|u_k(y)|^p}{|x-y|^{N-\alpha}}
 \,dx\,dy
+\int_{A_r}\int_{\mathbb{R}^N}\frac{|u_k(x)|^p|u_k(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy\\
&= \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|v_k(x)|^p|v_k(y)|^p}{|x-y|^{N-\alpha}}
 \,dx\,dy
+ \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|w_k(x)|^p|w_k(y)|^p}{|x-y|^{N-\alpha}}
 \,dx\,dy+O(\mu).
\end{aligned}
\end{equation}
Next, observe that for $k\geq 1$ large enough,  there exists $\epsilon'>0$ such that
\begin{equation}\label{compact8}
\int_{\mathbb{R}^N}|\nabla w_k|^2+V(x)|w_k|^2\geq\epsilon'.
\end{equation}
Indeed,  we can conclude from \eqref{compact1} and \eqref{compact4} 
that for $k>r+l$, it holds that
\begin{align*}
&\int_{\mathbb{R}^N}|\nabla w_k|^2+V(x)|w_k|^2 \\
&\geq \int_{B^c_{r+l}}|\nabla w_k|^2+(V_\infty-\mu)|w_k|^2\\
&=  \int_{|x|>k}|\nabla w_k|^2+(V_\infty-\mu)|w_k|^2
 +\int_{B_k\backslash{B_{r+l}}}|\nabla w_k|^2+(V_\infty-\mu)|w_k|^2\\
&\geq \int_{|x|>k}|\nabla u_k|^2+(V_\infty-\mu)|u_k|^2\\
&\geq \min\{1, V_\infty-\mu\}\epsilon_0 .
\end{align*}
Hence \eqref{compact8} holds.

Therefore, by \eqref{compact9} and \eqref{compact8}, the following equality 
and inequality hold.
\begin{gather}\label{compact10}
J(u_k)=J(v_k)+J(w_k)+O(\mu), \\
\label{compact11}
J(u_k)\geq J(w_k)+O(\mu), \\
\label{compact12}
J(u_k)-C\epsilon'\geq J(v_k)+O(\mu).
\end{gather}
\smallskip

\noindent\textbf{Step 4.}
Recall  $G(u)$ defined in Lemma~\ref{lem3.1}. By \eqref{compact9} and 
\eqref{compact13}, we deduce
\begin{equation}\label{compact14}
0=G(u_k)= G(v_k)+G(w_k)+O(\mu).
\end{equation}
 We shall complete the proof by  distinguishing three cases.
\smallskip

\noindent\textbf{Case 1.}
 Up to a subsequence, $G(v_k)\leq0$.
By Lemma~\ref{lem2.4} (i), for any $k\geq1$, there exists a unique  
$t_k>0$ such that $t_kv_k\in\mathcal{N}$. Then
 \begin{equation}\label{compact15}
 \int_{\mathbb{R}^N}|\nabla v_k|^2+V(x)|v_k|^2=t_k^{2p-2}\mathbb{D}(v_k).
 \end{equation}
Note that
 $$ 
\int_{\mathbb{R}^N}|\nabla v_k|^2+V(x)|v_k|^2\leq\mathbb{D}(v_k).
$$
This, combined with \eqref{compact15}, implies that $t_k\leq1$ uniformly.
By \eqref{compact12}, we obtain
\begin{equation}\label{compact16}
\begin{aligned}
c&\leq I(t_kv_k)=J(t_kv_k)\leq J(v_k) \\
&\leq J(u_k)-C\epsilon'+O(\mu)
=c-C\epsilon'+O(\mu)+o_{k}(1).
\end{aligned}
\end{equation}
Here and in the following part, we point out $o_k(1)\to 0$ as $k\to \infty$. 
By letting $\mu\to 0$ and $k\to \infty$, \eqref{compact16} yields a contradiction.
\smallskip

\noindent\textbf{Case 2.}
Up to a subsequence, $G(w_k)\leq 0$.
 For any $k\geq1$, there exists $s_k>0$ such that $s_kw_k\in \mathcal{N}$. 
Arguing as in Case 1, we have $s_k\leq 1$ uniformly. Define $\bar{w}_k=s_kw_k$. 
Then there exists $\theta_k>0$ such that 
$\theta_k\bar{w}_k\in \mathcal{N}_{V_\infty-\mu}$. By \eqref{compact4}, we have
\[
 \int_{\mathbb{R}^N}|\nabla \bar{w}_k|^2+(V_\infty-\mu)|\bar{w}_k|^2
\leq  \int_{\mathbb{R}^N}|\nabla \bar{w}_k|^2+V(x)|\bar{w}_k|^2
 = \mathbb{D}(\bar{w}_k),
\]
which implies that $\theta_k\leq1$ uniformly. Hence, by \eqref{compact11}, we deduce
\begin{equation}
\begin{aligned}
 c(V_\infty-\mu)
&\leq  I_{V_\infty-\mu}(\theta_k\bar{w}_k)\\
 &\leq  (\frac{1}{2}-\frac{1}{2p})\int_{\mathbb{R}^N}|\nabla \bar{w}_k|^2
 +(V_\infty-\mu)|\bar{w}_k|^2\\
&\leq  (\frac{1}{2}-\frac{1}{2p})\int_{\mathbb{R}^N}|\nabla \bar{w}_k|^2
 +V(x)|\bar{w}_k|^2\\
&\leq J(w_k)\\
 &\leq  J(u_k)+O(\mu)\\
 &= c+o_k(1)+O(\mu).
 \end{aligned}
\end{equation}
Letting $\mu\to 0$ and $k\to \infty$, we obtain a contradiction with \eqref{compact3}.
\smallskip

\noindent\textbf{Case 3.}
Up to a subsequence, $G(v_k)>0$ and $G(w_k)>0$.
According to \eqref{compact14}, we have
$$
G(w_k)= O(\mu)>0,\ G(v_k)= O(\mu)>0.
$$
For any $k\geq1$, there exists $s_k>0$ such that $s_kw_k\in \mathcal{N}$ and then 
$G(s_kw_k)=0$. So that
\begin{gather*}
\int_{\mathbb{R}^N}|\nabla w_k|^2+V(x)|w_k|^2=s_k^{2p-2}\mathbb{D}(w_k),\\
\int_{\mathbb{R}^N}|\nabla w_k|^2+V(x)|w_k|^2-\mathbb{D}(w_k)=O(\mu)>0,
\end{gather*}
which implies $s_k\geq1$ uniformly. Since $(w_k)_{k\geq1}$ is bounded,  
by \eqref{compact8}, we have
$$
s_k^{2p-2}=\frac{\int_{\mathbb{R}^N}|\nabla w_k|^2
+V(x)|w_k|^2}{\mathbb{D}(w_k)}\leq \frac{C}{\epsilon'-O(\mu)}.
$$
Hence $(s_k)_{k\geq 1}$ is uniformly bounded  when $\mu$ is small enough.


Now we need to distinguish two cases.
\smallskip

\noindent\textbf{Case 3-(i).}
Up to a subsequence, if $\lim_{k\to\infty} s_k=1$,  for $k$ large enough,
$1\leq s_k\leq 1+O(\mu)$. Using similar arguments as in Case 2, we have
\begin{equation}
\begin{aligned}
 c(V_\infty-\mu)&\leq  I_{V_\infty-\mu}(\theta_k\bar{w}_k)\\
 &\leq  (\frac{1}{2}-\frac{1}{2p})\int_{\mathbb{R}^N}|\nabla \bar{w}_k|^2+(V_\infty-\mu)|\bar{w}_k|^2\\
 &\leq  (\frac{1}{2}-\frac{1}{2p})\int_{\mathbb{R}^N}|\nabla \bar{w}_k|^2+V(x)|\bar{w}_k|^2\\
 &\leq (1+O(\mu))^2J(w_k)\\
 &\leq  (1+O(\mu))^2(J(u_k)+O(\mu))\\
 &= (1+O(\mu))^2(c+o_k(1)+O(\mu)).
 \end{aligned}
\end{equation}
Letting $\mu\to 0$ and $k\to \infty$, we obtain  a contradiction 
with \eqref{compact3}.
\smallskip

\noindent\textbf{Case 3-(ii).} Up to a subsequence, if $\lim_{k\to\infty} s_k=s_0>1$,
for $k$ large enough, $s_k>1$. On the other hand,  we have
\begin{align*}
O(\mu)=G(w_k)
&= \int_{\mathbb{R}^N}|\nabla {w}_k|^2+V(x)|{w}_k|^2-\mathbb{D}(w_k)\\
&= (1-s_k^{\frac{1}{2p-2}})\int_{\mathbb{R}^N}|\nabla {w}_k|^2+V(x)|{w}_k|^2.
\end{align*}
Hence
$$
\int_{\mathbb{R}^N}|\nabla {w}_k|^2+V(x)|{w}_k|^2=O(\mu),
$$
which  contradicts \eqref{compact8}.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 Since $(u_n)_{n\geq1}$ is uniformly bounded,  going if necessary to a subsequence, 
there exists $u_0\in H^1(\mathbb{R}^N)$
such that $u_n\rightharpoonup u_0$  in  $H^1(\mathbb{R}^N)$ and 
$u_n\to u_0$ a.e. in $\mathbb{R}^N$. By Lemma~\ref{lem3.1}, we have
 $I'(u_n)\to 0$ as $n\to\infty$,
and then $I'(u_0)=0$ because of \cite[Lemma 2.6]{wt}.

Now we  show $u_0\neq 0$. According to Lemma~\ref{lem3.2},
for any $\epsilon>0$, there exists $r>0$ such that, up to a subsequence,
$$
\|u_n\|_{H^1(B_r^c)}<\epsilon,\quad  \text{for any } n\geq1.
$$
Let $s\in[2,\frac{2N}{N-2})$. For  $n\geq1$ large enough, we have
\begin{align*}
|u_n-u_0|_{L^s(\mathbb{R}^N)}
&= |u_n-u_0|_{L^s(B_r)}+|u_n-u_0|_{L^s(B^c_r)}\\
&\leq \epsilon+C_0(\|u_n\|_{H^1(B_r^c)}+\|u_0\|_{H^1(B_r^c)})\\
&\leq (1+2C_0)\epsilon.
\end{align*}
Then we deduce
$u_n\to u_0$ in $L^s(\mathbb{R}^N)$ for any $s\in[2,\frac{2N}{N-2})$. 
This, combined with Br\'ezis-Lieb Lemma (see \cite[Theorem 1.32]{wm}), implies
$$
|u_n|^p\to|u_0|^p,\quad \text{in }  L^{\frac{2N}{N+\alpha}}(\mathbb{R}^N).
$$
Hence using \eqref{nonlocal}, we obtain
\begin{equation}\label{nonlocal1}
\begin{aligned}
&|\mathbb{D}(u_n)-\mathbb{D}(u_0)|\\
&\leq \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\big||u_n(x)|^p
 -|u_0(x)|^p\big||u_n(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy \\
&\quad +\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\big||u_n(y)|^p-|u_0(y)|^p\big|
 |u_0(x)|^p}{|x-y|^{N-\alpha}}\,dx\,dy.\\
&\leq C\big||u_n|^p-|u_0|^p \big|_{L^\frac{2N}{N+\alpha}}
 |u_n|^p_{L^\frac{2Np}{N+\alpha}}
+ C\big||u_n|^p-|u_0|^p \big|_{L^\frac{2N}{N+\alpha}}
 |u_0|^p_{L^\frac{2Np}{N+\alpha}}\\
&\to 0.
\end{aligned}
\end{equation}
Note that $I'(u_n)u_n=0$ and $I'(u_0)u_0=0$. Then
\begin{gather*}
I(u_n)=(\frac{1}{2}-\frac{1}{2p})\mathbb{D}(u_n),\\
I(u_0)=(\frac{1}{2}-\frac{1}{2p})\mathbb{D}(u_0).
\end{gather*}
Since $I(u_n)\to c$ as $n\to \infty$,
we conclude from \eqref{nonlocal1} that $I(u_n)\to I(u_0)=c$.
Therefore, $u_0$ is a ground state solution of \eqref{model-1}. 
This completes the proof. 
\end{proof} 

\subsection*{Acknowledgments}
Research was supported partially by the
National Natural Science Foundation of China (Grant No.
11571371)

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