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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 03, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/03\hfil Solutions for Choquard type equations]
{Existence and nonexistence of nontrivial solutions for Choquard type equations}

\author[T. Wang \hfil EJDE-2016/03\hfilneg]
{Tao Wang}

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

\thanks{Submitted November 8, 2015. Published January 4, 2016.}
\subjclass[2010]{35A15, 35J20, 35J60}
\keywords{Choquard equation; nonlocal nonlinearities; variational methods}

\begin{abstract}
 In this article, we consider the nonlocal problem
 $$
 -\Delta u+u=q(x)\Big(\int_{\mathbb{R}^N}\frac{q(y)|u(y)|^p}{|x-y|^{N-\alpha}}dy
 \Big)|u|^{p-2}u,\quad x\in \mathbb{R}^N,
 $$
 where $N\geq3$, $\alpha\in (0,N)$, $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2}$
 and $q(x)$ is a given potential. Under suitable assumptions on $q(x)$,
 we prove the existence and nonexistence of nontrivial solutions.
\end{abstract}

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



\section{Introduction}

In this article, we are concerned with the equation
\begin{equation}\label{model-1}
-\Delta u+u=q(x)\Big(\int_{\mathbb{R}^N}\frac{q(y)|u(y)|^p}{|x-y|^{N-\alpha}}dy
\Big)|u|^{p-2}u,\quad  x\in \mathbb{R}^N,
\end{equation}
where $N\geq3$, $\alpha\in(0,N)$, $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2}$,
$q(x)\geq 0$ and $q(x)$ is continuous  in $\mathbb{R}^N$.

It is well known that   when $N=3$, $\alpha=2$, $p=2$ and $q\equiv1$, Equation
\eqref{model-1}  becomes the classical stationary Choquard equation
 \begin{equation}\label{model-10}
-\Delta u+u=(|u|^2\ast{\frac{1}{|x|}})u,\quad\text{in } \mathbb{R}^3.
\end{equation}
It 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{LEH}.
In 1996, Penrose proposed \eqref{model-10} as a model of self-gravitating matter,
in a program in which quantum state reduction is understood as a gravitational
phenomenon \cite{mgp},  see also  \cite{GJ, LEHS, pr} for more details.

In 1977, using symmetric rearrangement inequalities,
Lieb \cite{LEH} showed the existence and uniqueness of the minimizer of
\eqref{model-10} up to translations. Later, Lions \cite{LPL} proved
the existence of a sequence of radially symmetric solutions
to \eqref{model-10} by dual variational methods.
Further results for related problems can be founded in
\cite{an, cms, pl, mgo, nm, tm, ww} and references therein.
In recent years,  the existence and  properties of solutions for the
 generalized Choquard type equation \eqref{model-1} with $q\equiv1$ have
been considered by many authors. In 2010,  Ma and Zhao  \cite{MZ}
proved the positive solutions for the generalized Choquard equation
 \eqref{model-1} with $q\equiv1$ must be radially symmetric and monotone
decreasing about some point under
appropriate assumptions on $p,\alpha, N$, see also \cite{ccs, csv}.
They also showed the positive solutions  of \eqref{model-10} is uniquely determined,
up to translations.
Moroz and Van Schaftingen \cite{vv} obtained the existence,  regularity,
positivity and radial symmetry of ground state solution  of
\eqref{model-1} with $q\equiv 1$  for the optimal range of parameters.
They also derived  the sharp decay asymptotic of the ground state solution,
see also \cite{vj}.
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),\quad \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 primate
function of $f(s)$ and $\epsilon$ is a positive parameter,
see  \cite{acoy, css, vjv} for more related problems.

 Motivated by  the above work, in this article,  we consider  \eqref{model-1}
with the nonlinear potential $q$ on the right side of the equation.
To be precise, using variational method, we investigate the existence and
nonexistence of nontrivial solutions to  \eqref{model-1} where  nonlinear
 potential $q$ is radial or non-radial.
To deduce our statements, we need the following assumptions:
\begin{itemize}
\item[(H1)] $\lim_{|x|\to \infty}q(x)= q_\infty$,
where $q_{\infty}>0$ is a  positive number;

\item[(H2)]  $\lim_{|x|\to \infty}q(x)= 0$;

\item[(H3)]  $q$ is bounded in $\mathbb{R}^N$ and there exists $R_0>0$ such that
$\min_{2R\leq|x|\leq4R}q(x)\geq\max_{|x|\leq R}q(x)$ for all $R\geq R_0$;

\item[(H4)] $q$ is radial in $\mathbb{R}^N$ and $q(r)\leq C(1+r^l)$ with
$0\leq l<\frac{(N-1)p-N-\alpha}{2}$,
where $C>0$ is a positive constant.
\end{itemize}
We remark (H1)--(H4) were introduced by Ding and Ni \cite{dingni} with some
 modifications.
Recall here 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.
Now we are ready to state our main results.

\begin{theorem}\label{minmax2}
Let  $N\geq3$, $\alpha\in(0,N)$, $p\in[2,\frac{N+\alpha}{N-2})$
and $q$ satisfies {\rm (H1)}. Then the following statements are true:
\begin{itemize}
\item[(i)] If $\lim_{|x|\to \infty}q(x)= \inf_{x\in \mathbb{R}^N}q(x)$,
 then \eqref{model-1} has a ground state solution in $H^1(\mathbb{R}^N)$.

\item[(ii)] If $\lim_{|x|\to\infty}q(x)=\sup_{x\in\mathbb{R}^N}q(x)$ and
$q$ is not constant, then \eqref{model-1} has no  ground state solution
in $H^1(\mathbb{R}^N)$.
\end{itemize}
\end{theorem}

Note that if $\alpha\leq N-4$, then assumptions of Theorem \ref{minmax2} can
not be satisfied.

\begin{theorem}\label{minmax25}
Let $N\geq3$, $\alpha\in(0,N)$,  $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2})$.
Then the following statements are true:
\begin{itemize}
\item[(i)] If $q$ satisfies (H2)  and $q\not\equiv 0$.
Then \eqref{model-1} has a ground state solution in $H^1(\mathbb{R}^N)$.

\item[(ii)] Suppose $q$ satisfies (H3) and $q$ is not constant.
Then  \eqref{model-1} has no   ground state solution in $H^1(\mathbb{R}^N)$.
\end{itemize}
\end{theorem}

\begin{theorem}\label{minmax21}
Let $N\geq3$, $\alpha\in(0,N)$,  $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2})$.
If $q$ satisfies {\rm (H4)} and $q\not\equiv 0$,  then \eqref{model-1}
has a  nonnegative radial   solution in $H_r^1(\mathbb{R}^N)$ defined in section 2.
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
If the nontrivial radial solution $u$ obtained in Theorem \ref{minmax21}
tends to zero exponentially fast at infinity, then according to the proof
of symmetric criticality principle (see Theorem 1.28 in\cite{wm}), we conclude
$u$ is a  solution of \eqref{model-1} in $H^1(\mathbb{R}^N)$.
\end{remark}

\begin{theorem}\label{minmax26}
Let $N\geq3$, $\alpha\in(0,N)$,  $p\in (\frac{N+\alpha}{N-1},\frac{N+\alpha}{N-2})$.
Suppose that  $q$ satisfies {\rm (H4)} and  $q(r)\to +\infty$ as $r\to\infty$.
Then for large $k$,   \eqref{minmax22} has two nontrivial weak solutions
in $H^1_0(B_k)$ defined in section 2, one of which is radial while the other is not.
\end{theorem}

The remainder of this article is organized as follows.
In section 2, we introduce some notation and give some important compactness lemmas,
which play a key role for the  the existence results.
In  sections 3 and 4, we prove our main results.
Throughout the paper,  we write $C>0$ for different positive constant.

\section{Preliminary results}\label{Sec2}

 We shall use the following  notation:\\
$\bullet$ Let $N$ and $k$ be positive integers and $B_R(0)$
 be an open  ball of radius $R$  centered at the origin in $\mathbb{R}^N$.\\
$\bullet$ $H^1(\mathbb{R}^N)$ is the usual Sobolev space with the standard norm
$$
\|u\|=\Big(\int_{\mathbb{R}^N}(|\nabla u|^2+|u|^2)dx\Big)^{1/2}.
$$
$H^1_{r}(\mathbb{R}^N)$ is the set of all radial functions in $H^1(\mathbb{R}^N)$.\\
$\bullet$ Let $C_c^\infty({B_k(0)})$ be the set of infinitely differential
functions with compact support in $B_k({0})$ and $H^1_0(B_k)$ be the closure
of $C_c^\infty({B_k(0)})$ in the norm defined by
 $$
\|u\|_{H^1_0(B_k)}=\Big(\int_{B_k(0)}(|\nabla u|^2+|u|^2)dx\Big)^{1/2}.
$$
$H^1_{0,r}(B_k)$ is the set of all radial functions in $H^1_{0}(B_k)$.
We can identify $u\in H_0^1(B_k)$ with its extension to $\mathbb{R}^N$
 obtained by setting $u=0$ in $\mathbb{R}^N\backslash B_k$.\\
$\bullet$ 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 $|u|_{L^s}=|u|_{L^s(\Omega)}$.\\
$\bullet$ The dual space of $H^1(\mathbb{R}^N)$ is denoted by $H^{-1}(\mathbb{R}^N)$.\\
$\bullet$  Let $\langle \cdot,\cdot\rangle$ be the duality pairing between
$H^1(\mathbb{R}^N)$ and $H^{-1}(\mathbb{R}^N)$.\\
From this, the  energy functional $I: H^1(\mathbb{R}^N)\to \mathbb{R}$ associated
with \eqref{model-1} is defined  by
 $$
I(u)=\frac{1}{2}\|u\|^2-
 \frac{1}{2p}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{q(x)q(y)|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy.
$$
Suppose $q$ is bounded in $\mathbb{R}^N$. Then the functional is well
defined by Hardy-Littlewood-Sobolev inequality (see\cite{llm}),
 which states that if $\frac{N+\alpha}{N}\leq p\leq\frac{N+\alpha}{N-2}$
and $u\in L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$, then
\begin{equation}\label{model-19}
\begin{aligned}
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
 \frac{q(x)q(y)|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy
&\leq  C(N,\alpha, s,t)|q|^2_{L^\infty}|u^{p}|_{L^s}|u^{p}|_{L^t}\\
&= C(N,\alpha,s,t)|q|^2_{L^\infty}|u|^{p}_{L^{sp}}|u|^{p}_{L^{tp}}
< \infty.
\end{aligned}
\end{equation}
where $C$  depends only on $N,\alpha,s,t$, and
$\frac{1}{s}+\frac{1}{t}+\frac{N-\alpha}{N}=2$.  This also implies that
$I$ is $C^1$ functional whose derivative is given by
$$
\langle I'(u), v\rangle=\int_{\mathbb{R}^N}(\nabla u \nabla v+uv)dx
-\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
 \frac{q(x)q(y)|u(y)|^p|u(x)|^{p-2}u(x)v(x)}{|x-y|^{N-\alpha}}\,dx\,dy
$$
for all $v\in H^1(\mathbb{R}^N)$.
It is easy to see  the critical points of $I$ are solutions to \eqref{model-1}
in the weak sense. We consider the Nehari manifold
$$
\mathcal{N}=\{u\in H^1(\mathbb{R}^N)\backslash\{0\}:\langle I'(u),u\rangle=0\}$$
and $c=\inf_{u\in\mathcal{N}} I(u)$.

In what follows, we consider the limit problem when $q$ satisfies (H1)
 \begin{equation}\label{model-2}
-\Delta u+u=(\int_{\mathbb{R}^N}
\frac{q_\infty^2|u(y)|^p}{|x-y|^{N-\alpha}}dy)|u|^{p-2}u,\quad x\in \mathbb{R}^N.
\end{equation}
The associated  energy functional is
$$
I_{\infty}(u)=\frac{1}{2}\|u\|^2
-\frac{1}{2p}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{q_\infty^2|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy,
$$
 and the corresponding Nehari manifold is
$$
\mathcal{N}_\infty=\{u\in H^1(\mathbb{R}^N)\backslash\{0\}:
\langle I_{\infty}'(u),u\rangle=0\}.
$$
We define $c_{\infty}=\inf_{u\in\mathcal{N}_\infty} I_\infty(u)$.

 For convenience, we introduce
$$
\mathbb{D}(u)=\int_{\mathbb{R}^N}
\int_{\mathbb{R}^N}\frac{q(x)q(y)|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy,\quad
 J(u)=\frac{\|u\|^2}{\mathbb{D}^{1/p}(u)}.
$$
The existence of ground state solution for \eqref{model-2}
has been  investigated in \cite[Theorem 1]{vv}.

\begin{lemma}[\cite{vv}]\label{lemm2.1}
Let $N\geq3$, $\alpha\in(0,N)$,  $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2})$.
Then there exists a ground state solution $w\in H^1(\mathbb{R}^N)$ such that
$w$ satisfies \eqref{model-2} weakly in $\mathbb{R}^N$ and
$c_\infty=I_\infty(w)$.
\end{lemma}

In the sequel, we shall  establish a compactness lemma which plays an important
role in our existence results. To achieve this, we need some basic lemmas.

\begin{lemma}[\cite{vv}] \label{minmax3}
Let $\Omega\subset \mathbb{R}^N$ be a domain, $s\in [1,\infty)$ and
$(u_n)_{n\geq1}$ be  a bounded sequence in $L^r(\Omega)$.
If $u_n\to u$ almost everywhere on $\Omega$
 as $n\to\infty$, then for every $s\in[1,r]$,
 \begin{displaymath}
 \lim_{n\to\infty}\int_{\Omega}\big||u_n|^s-|u_n-u|^s-|u|^s\big|^{r/s}=0.
 \end{displaymath}
\end{lemma}

\begin{lemma}[\cite{wim}]\label{minmax4}
Let $\Omega\subset \mathbb{R}^N$ be a domain, $s\in (1,\infty)$ and $(u_n)_{n\geq1}$
be  a bounded sequence in $L^s(\Omega)$. If $u_n\to u$ almost everywhere on $\Omega$
 as $n\to\infty$, then $u_n\rightharpoonup u$ weakly in $L^s(\Omega)$.
\end{lemma}

 According to Lemmas \ref{minmax3} and \ref{minmax4}, we obtain the following
three lemmas, whose proofs are similar as that of
\cite[Proposition A.1]{sd} and \cite[Lemma 2.15]{lgb}
with some necessary modifications. For the sake of completeness,
we prove them  here. In addition,  we remark that  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 handle with the nonlocal problem.

\begin{lemma}\label{minmax5}
Let $N\geq3$, $\alpha\in(0,N)$,  $p\in (\frac{N+\alpha}{N},\frac{N+\alpha}{N-2})$
and suppose  $q$ is bounded in $\mathbb{R^N}$. If $(u_n)_{n\geq1}$
is   a bounded sequence in $L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$, and
$u_n\to u$ almost everywhere on $\mathbb{R}^N$
 as $n\to\infty$, then
 $$
\lim_{n\to\infty}(\mathbb{D}(u_n)-\mathbb{D}(u_n-u))=\mathbb{D}(u).
$$
\end{lemma}

\begin{proof}
The proof can be split into  three steps.
\smallskip

\noindent\textbf{Step 1.} For every $n$, we have
%\label{minmax6}
\begin{align*}
&\mathbb{D}(u_n)-\mathbb{D}(u_n-u) \\
&= \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{q(x)
 q(y)(|u_n(y)|^p-|u_n-u|^p(y))(|u_n(x)|^p-|u_n-u|^p(x))}{|x-y|^{N-\alpha}}
\,dx\,dy\\
& \quad +2\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{q(x)
 q(y)(|u_n(y)|^p-|u_n-u|^p(y))|u_n-u|^p(x)}{|x-y|^{N-\alpha}}\,dx\,dy\\
&= :I_1+2I_2.
\end{align*}

\noindent\textbf{Step 2.} By Lemmas \ref{minmax3} and \ref{minmax4},
the following statements are true,  as $n\to\infty$,
\begin{gather}\label{minmax27}
q(|u_n|^p-|u_n-u|^p)\to q|u|^p\quad \text{strongly in }
  L^{\frac{2N}{N+\alpha}}(\mathbb{R}^N),
\\
\label{minmax28}
\begin{gathered}
\int_{\mathbb{R}^N}\frac{|u_n(y)|^p-|u_n-u|^p(y)}{|x-y|^{N-\alpha}}dy\to
 \int_{\mathbb{R}^N}\frac{|u(y)|^p}{|x-y|^{N-\alpha}}dy\\
\text{strongly in }  L^{\frac{2N}{N-\alpha}}(\mathbb{R}^N),
\end{gathered}
\\
\label{minmax29}
q|u_n-u|^p\rightharpoonup 0\quad  \text{weakly in }
  L^{\frac{2N}{N+\alpha}}(\mathbb{R}^N).
 \end{gather}

\noindent\textbf{Step 3.}
 By \eqref{model-19} and \eqref{minmax27}, we obtain
 \begin{equation}\label{minmax7}
 \begin{aligned}
 &|I_1-\mathbb{D}(u)|\\
&\leq |\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
 \Big(q(x)q(y)(|u_n(y)|^p-|u_n-u|^p(y)-|u(y)|^p)(|u_n(x)|^p\\
&\quad -|u_n-u|^p(x))\Big)\big/|x-y|^{N-\alpha} \,dx\,dy|\\
&\quad +|\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
 \frac{q(x)q(y)|u(y)|^p(|u_n(x)|^p-|u_n-u|^p(x)-|u(x)|^p)}{|x-y|^{N-\alpha}}\,dx\,dy|
\\
 &\leq  C|q|_{L^\infty}\left|q|u_n|^p-q|u_n-u|^p-q|u|^p\right|_{L^{\frac{2N}{N+\alpha}}}
 \left||u_n|^p-|u_n-u|^p\right|_{L^{\frac{2N}{N+\alpha}}}\\
 &\quad +C|q|_{L^\infty}\left|q|u_n|^p-q|u_n-u|^p-q|u|^p
 \right|_{L^{\frac{2N}{N+\alpha}}}
 |u|^p_{L^{\frac{2Np}{N+\alpha}}}
 \to 0,
\end{aligned}
 \end{equation}
as $n\to \infty$.
By \eqref{model-19},\eqref{minmax27}, \eqref{minmax28} and \eqref{minmax29},
we conclude that
%\label{minmax9}
\begin{align*}
|I_2|
&\leq |\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{q(x)q(y)(|u_n(y)|^p
-|u_n-u|^p(y)-|u(y)|^p)|u_n-u|^p(x)}{|x-y|^{N-\alpha}}\,dx\,dy|\\
&\quad +|\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{q(x)q(y)|u(y)|^p
 |u_n-u|^p(x)}{|x-y|^{N-\alpha}}\,dx\,dy|
\to 0,
\end{align*}
as $n\to \infty$.
Then  the proof is complete.
\end{proof}

\begin{lemma}\label{minmax20}
Let $N\geq3$, $\alpha\in(0,N)$,  $p\in [2,\frac{N+\alpha}{N-2})$ and suppose
$q$ is bounded in $\mathbb{R^N}$. If $(u_n)_{n\geq1}$
is   a bounded sequence in $L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$, and
$u_n\to u$ almost everywhere on $\mathbb{R}^N$
 as $n\to\infty$, then
 $$
\lim_{n\to\infty}(\mathbb{D'}(u_n)-\mathbb{D'}(u_n-u))
=\mathbb{D'}(u)\quad \text{in } H^{-1}(\mathbb{R}^N).
$$
\end{lemma}

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

\noindent\textbf{Step 1.} By Lemmas \ref{minmax3} and  \ref{minmax4},
for any $v\in  H^{1}(\mathbb{R}^N)$, we have
\begin{equation}\label{minmax10}
q|u_n|^{p-2}u_n-q|u_n-u|^{p-2}(u_n-u)\to q|u|^{p-2}u \quad
\text{strongly in }  L^{\frac{2Np}{(N+\alpha)(p-1)}}(\mathbb{R}^N).
\end{equation}
\smallskip

\noindent\textbf{Step 2.} It is easy to check that
\begin{equation}
\begin{aligned}
&\langle\mathbb{D}'(u_n),v\rangle-\langle\mathbb{D}'(u_n-u),v\rangle\\
&= 2p\Big[\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{q(x)q(y)|u_n(y)|^p|u_n(x)|^{p-2}u_n(x)v(x)}{|x-y|^{N-\alpha}}\,dx\,dy\\
& \quad -\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{q(x)q(y)|u_n-u|^p(y)|u_n-u|^{p-2}(x)(u_n-u)(x)v(x)}{|x-y|^{N-\alpha}}
\,dx\,dy\Big]\\
&=: 2pK.
\end{aligned}
\end{equation}
and
\begin{align*}
K&= \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\Big(q(x)q(y)(|u_n(y)|^p-|u_n-u|^p(y))(|u_n(x)|^{p-2}u_n(x)v(x)\\
&\quad - |u_n-u|^{p-2}(x)(u_n-u)(x)v(x))\Big)\big/ |x-y|^{N-\alpha}\,dx\,dy
\\
&\quad +\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\Big(q(x)q(y)(|u_n(y)|^{p}-|u_n-u|^{p}(y))|u_n-u|^{p-2}(x)(u_n-u)(x)v(x)\Big)\\
&\quad\div |x-y|^{N-\alpha}\,dx\,dy\\
&\quad +\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\Big(q(x)q(y)|u_n-u|^{p}(y)(|u_n(x)|^{p-2}u_n(x)v(x)\\
&\quad -|u_n-u|^{p-2}(x)(u_n-u)(x)v(x))
\Big)\big/ |x-y|^{N-\alpha}\,dx\,dy\\
&=: K_1+K_2+K_3.
\end{align*}

\noindent\textbf{Step 3.} By direct calculations, from \eqref{model-19},
\eqref{minmax27} and \eqref{minmax10}  we  deduce that for  $n$ large enough,
\begin{align*}
&|K_1-\frac{1}{2p}\langle\mathbb{D}'(u),v\rangle|\\
&=\Big|\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\Big(q(x)q(y)(|u_n(y)|^p-|u_n-u|^p(y)-|u(y)|^p)(|u_n(x)|^{p-2}u_n(x)v(x)\\
&\quad - |u_n-u|^{p-2}(x)(u_n-u)(x)v(x))\Big)\big/ |x-y|^{N-\alpha}\,dx\,dy
\\
&\quad +\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\Big(q(x)q(y)|u(y)|^p(|u_n(x)|^{p-2}u_n(x)v(x)\\
&\quad -|u_n-u|^{p-2}(x)(u_n-u)(x)v(x))-|u(x)|^{p-2}u(x)v(x)\Big)
\big/|x-y|^{N-\alpha}\,dx\,dy|\\
&\leq C|q|_{L^\infty}|q|u_n|^p-q|u_n-u|^p-q|u|^p|_{L^{\frac{2N}{N+\alpha}}}
 \big||u_n|^{p-2}u_n\\
&\quad -|u_n-u|^{p-2}(u_n-u)\big|_{L^{\frac{2Np}{(N+\alpha)(p-1)}}}\|v\|\\
&\quad+C|q|_{L^\infty}|u|^p_{L^{\frac{2Np}{N+\alpha}}}
\big||q|u_n|^{p-2}u_n-q|u_n-u|^{p-2}(u_n-u) \\
&\quad -q|u|^{p-2}  u\big|_{L^{\frac{2Np}{(N+\alpha)(p-1)}}} \|v\|
=o(1)\|v\|.
\end{align*}
Here and in the following part,  we point out that $o(1)\to 0$ as $n\to \infty$.

Since $u\in L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$,
for any $\epsilon>0$, there exists $R_1>0$ such that
\[
|u|_{L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N\backslash B_{R_1}(0))}<\epsilon.
\]
Fix $R_1>0$. Then there exists $R_2>0$  large enough such that
\begin{align}
&\big|\int_{\mathbb{R}^N\backslash B_{R_1+R_2}(0)}\int_{B_{R_1}(0)}
\frac{q(x)q(y)|u(y)|^p|u_n-u|^{p-2}(x)(u_n-u)(x)v(x)}{|x-y|^{N-\alpha}}\,dx\,dy\big|
\nonumber \\
&\leq C{R_2}^{\alpha-N}\|v\|
<\epsilon\|v\| \label{minmax40}
\end{align}
 Note that $|u_n-u|^p\to 0 $ in $L_{\rm loc}^{\frac{2N}{N+\alpha}}(\mathbb{R}^N)$.
For $n$ large enough, we deduce from \eqref{model-19}, \eqref{minmax27}
and  \eqref{minmax40} that
\begin{align*}
&|K_2|\\
&\leq \big|\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\Big(q(x)q(y)(|u_n(y)|^{p}
 -|u_n-u|^{p}(y)-|u(y)|^{p})|u_n-u|^{p-2}(x)\\
&\quad\times (u_n-u)(x)v(x)\Big)\big/|x-y|^{N-\alpha}
 \,dx\,dy\big|\\
&\quad +\big|\int_{\mathbb{R}^N}\int_{\mathbb{R}^N\backslash B_{R_1}(0)}
 \frac{q(x)q(y)|u(y)|^p|u_n-u|^{p-2}(x)(u_n-u)(x)v(x)}{|x-y|^{N-\alpha}}\,dx\,dy\big|\\
&\quad + |\int_{\mathbb{R}^N\backslash B_{R_1+R_2}(0)}\int_{B_{R_1}(0)}
 \frac{q(x)q(y)|u(y)|^p|u_n-u|^{p-2}(x)(u_n-u)(x)v(x)}{|x-y|^{N-\alpha}}\,dx\,dy\big|\\
&\quad +\big|\int_{B_{R_1+R_2}(0)}\int_{B_{R_1}(0)}\frac{q(x)q(y)|u(y)|^p
 |u_n-u|^{p-2}(x)(u_n-u)(x)v(x)}{|x-y|^{N-\alpha}}\,dx\,dy\big|\\
&=o(1)\|v\|.
\end{align*}
Similarly, $K_3=o(1)\|v\|$ for $n$ large enough.
This completes the proof.
\end{proof}

\begin{lemma}\label{minmax18}
Let $N\geq3$, $\alpha\in(0,N)$,  $p\in [2,\frac{N+\alpha}{N-2})$ and suppose
 $q$ is bounded in $\mathbb{R^N}$. If $(u_n)_{n\geq1}$ is a sequence such
that $u_n\rightharpoonup u$ weakly in $H^1(\mathbb{R}^N)$, then
$\langle\mathbb{D}'(u_n),v\rangle\to \langle\mathbb{D}'(u),v\rangle$
for all $v\in H^1(\mathbb{R}^N)$.
\end{lemma}

\begin{proof}
Since $u_n\rightharpoonup u$ weakly in $H^1(\mathbb{R}^N)$, $(u_n)_{n\geq1}$
is bounded in $H^1(\mathbb{R}^N)$. Going if necessary to a subsequence,
 we assume $u_n\to u$ a.e. on $\mathbb{R}^N$.
For any $v\in H^1(\mathbb{R}^N)$, it is easy to verify that
\begin{equation}\label{minmax12}
|u_n|^{p-2}u_nv\to |u|^{p-2}uv\quad  \text{strongly in }
 L^{\frac{2N}{N+\alpha}}(\mathbb{R}^N).
\end{equation}

\noindent\textbf{Step 1.} A direct calculation yields
\begin{align*}
\langle\mathbb{D}'(u_n),v\rangle-\langle\mathbb{D}'(u),v\rangle
&= 2p[\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{q(x)q(y)|u_n(y)|^p|u_n(x)|^{p-2}u_n(x)v(x)}{|x-y|^{N-\alpha}}\,dx\,dy\\
&\quad-\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{q(x)q(y)|u(y)|^p|u(x)|^{p-2}u(x)v(x)}{|x-y|^{N-\alpha}}\,dx\,dy]\\
&=:2pT.
\end{align*}
and
\begin{align}
T&= \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{q(x)q(y)(|u_n(y)|^p-|u(y)|^p)|u_n(x)|^{p-2}u_n(x)v(x)}{|x-y|^{N-\alpha}}
 \,dx\,dy \nonumber \\
& \quad +\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{q(x)q(y)|u(y)|^p(|u_n(x)|^{p-2}u_n(x)v(x)
 -|u(x)|^{p-2}u(x)v(x))}{|x-y|^{N-\alpha}}\,dx\,dy \nonumber \\
&= T_{1}+T_{2} \label{minmax31}
\end{align}


\noindent\textbf{Step 2.}
 Since $v\in L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$, for any $\epsilon>0$,
there exists $R_1>0$ such that
$|v|_{L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N\backslash B_{R_1}(0))}<\epsilon$.
Fix $R_1>0$. Then there exists $R_2>0$  large enough such that
\begin{align}
&\big|\int_{B_{R_1}(0)}\int_{\mathbb{R}^N\backslash B_{R_1+R_2}(0)}
\frac{q(x)q(y)(|u_n(y)|^p-|u(y)|^p)|u_n(x)|^{p-2}u_n(x)v(x)}{|x-y|^{N-\alpha}}
\,dx\,dy\big| \nonumber\\
&\leq C{R_2}^{\alpha-N} <\epsilon. \label{minmax32}
\end{align}
  Note that $|u_n|^p\to |u|^p$ in $L_{\rm loc}^{\frac{2N}{N+\alpha}}(\mathbb{R}^N)$.
Letting $n\to \infty$ and then $\epsilon\to 0$, we conclude from
  \eqref{model-19}, \eqref{minmax31} and \eqref{minmax32} that
\begin{align*}
&|T_{1}| \\
&\leq \big|\int_{B_{R_1}(0)}\int_{B_{R_1+R_2}(0)}
\frac{q(x)q(y)(|u_n(y)|^p-|u(y)|^p)|u_n(x)|^{p-2}u_n(x)v(x)}
 {|x-y|^{N-\alpha}}\,dx\,dy\big|\\
&\quad+\big|\int_{B_{R_1}(0)}\int_{\mathbb{R}^N\backslash B_{R_1+R_2}(0)}
\!\!\frac{q(x)q(y)(|u_n(y)|^p-|u(y)|^p)|u_n(x)|^{p-2}u_n(x)v(x)}
 {|x-y|^{N-\alpha}}\,dx\,dy\big|\\
&\quad +\big|\int_{\mathbb{R}^N\backslash B_{R_1}(0)}\int_{\mathbb{R}^N}
\frac{q(x)q(y)(|u_n(y)|^p-|u(y)|^p)|u_n(x)|^{p-2}u_n(x)v(x)}
 {|x-y|^{N-\alpha}}\,dx\,dy\big|
\to 0.
\end{align*}
On the other hand, it follows from \eqref{minmax12} that $T_{2}\to 0$ as
$n\to \infty$.
This completes the proof.
\end{proof}

Now, we are ready to prove the compactness lemma,  following
exactly the same lines as the proof of \cite[Proposition 8.4]{wm}.

\begin{definition} \label{def2.1} \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,\quad  I'(u_n)\to 0.
\end{equation}
\end{definition}

\begin{lemma} \label{minmax16}
Let  $N\geq3$, $\alpha\in(0,N)$ and $2\leq p<\frac{N+\alpha}{N-2}$.
Suppose $q$ satisfies (H1) and $(u_n)_{n\geq1}\subset H^1(\mathbb{R}^N)$ is
a $(PS)_C$ sequence of $I$.
Then, replacing $(u_n)_{n\geq1}$ if necessary by a subsequence, there
exists a solution $v_0\in H^1(\mathbb{R}^N)$
of  \eqref{model-1},   $\{v_1,v_2,\cdot\cdot\cdot, v_k\}\subset H^1(\mathbb{R}^N)$
of solutions of \eqref{model-2}, and $k$ sequences
$(y_{n}^{j})_{n\geq1}$, $1\leq j\leq k$ satisfying
\begin{gather*}
|y_{n}^{j}|\to \infty, \quad |y_{n}^{j}-y_{n}^{j'}|\to \infty,\quad
j\neq j',\; n\to \infty,\\
\|u_n-v_0-\sum_{j=1}^{k}v_j(\cdot-y_{n}^{j})\|\to 0,\\
\|u_n\|^2\to\sum_{j=0}^{k}\|v_j\|^2,\\
I(v_0)+\sum_{j=1}^{k}I_{\infty}(v_j)=C.
\end{gather*}
\end{lemma}

\begin{proof}
The proof can be split into three steps.
\smallskip

\noindent\textbf{Step 1.}
Since $I(u_n)\to C$ and $ I'(u_n)\to 0$, then for $n$ large enough, we have
\begin{align*}
C+1+\|u_n\|
&\geq  I(u_n)-\frac{1}{2p}\langle I'(u_n),u_n\rangle\\
&= (\frac{1}{2}-\frac{1}{2p})\int_{\mathbb{R}^N}(|\nabla u_n|^2+|u_n|^2)\\
&= (\frac{1}{2}-\frac{1}{2p})\|u_n\|^2,
\end{align*}
which yields that $\|u_n\|$ is bounded.
\smallskip

\noindent\textbf{Step 2.} We assume that $u_n\rightharpoonup v_0$
in $H^1(\mathbb{R}^N)$ and $u_n\to v_0$ a.e. on $\mathbb{R}^N$.
Then we claim that
$I'(v_0)=0$ and $u_{n}^{1}:=u_n-v_0$  such that
\begin{gather*}
\|u_{n}^{1}\|^2=\|u_n\|^2-\|v_0\|^2+o(1),\\
 I_{\infty}(u_{n}^{1})\to C-I(v_0),\\
I_{\infty}'(u_{n}^{1})\to 0 \quad\text{in } H^{-1}(\mathbb{R}^N).
\end{gather*}
Indeed, applying Lemma \ref{minmax18}, we have $I'(v_0)=0$.
Since $\lim_{|x|\to \infty}q(x)=q_\infty$ and
$u_{n}^{1}\to 0$ in $L_{\rm loc}^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$,
then we derive from  \eqref{model-19} that for $n$ large enough,
\begin{equation}{\label{model-7}}
I_{\infty}(u_{n}^{1})=I(u_{n}^{1})+o(1).
\end{equation}
On the other hand, $\|u_{n}^{1}\|^2=\|u_n\|^2-\|v_0\|^2+o(1)$.
Then it follows from  Lemma \ref{minmax5} and \eqref{model-7} that
\[
I_{\infty}(u_{n}^{1})
= I(u_n)-I(v_0)+o(1)
= C-I(v_0)+o(1).
\]
Since  $I'(u_n)\to 0$ in $H^{-1}(\mathbb{R}^N)$, it follows from
Lemma \ref{minmax20} that for $n$ large enough,
\begin{equation}{\label{model-12}}
\begin{aligned}
 I'_{\infty}(u_{n}^{1})
&= I'(u_{n}^{1}) +o(1)\\
&= I'(u_n)-I'(v_0)+o(1)
= o(1).
\end{aligned}
\end{equation}
Therefore, the claim holds.
\smallskip

\noindent\textbf{Step 3.} Let
$$
\delta:=\limsup_{n\to\infty}\Big(\sup_{y\in \mathbb{R}^N}\int_{B_1(y)}|u_{n}^{1}|^2dx
\Big).
$$
If $\delta=0$, by Lemma 1.21 in \cite{wm}, we have $u_{n}^{1}\to 0$ in
 $L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$. Moreover,
$I'_{\infty}(u_{n}^{1})\to 0$, then it follows from \eqref{model-19} that,
for $n$ large enough,
\[
\|u_{n}^{1}\|^2=\langle I'_{\infty}(u_{n}^{1}),u_{n}^{1}\rangle
+\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{q_\infty^2|u_{n}^{1}
(y)|^p|u_{n}^{1}(x)|^p}{|x-y|^{N-\alpha}}\,dx\,dy
=o(1),
\]
and then we complete our proof.
If $\delta>0$, then there exists a sequence $(y_{n}^{1})_{n\geq1}$ such that
$ \int_{B_1(y_{n}^{1})}|u_{n}^{1}|^2>\frac{\delta}{2}$.
Let $v_{n}^{1}:=u_{n}^{1}(\cdot+y_{n}^{1})$.
Then $v_{n}^{1}\rightharpoonup v_1$ weakly in $H^1(\mathbb{R}^N)$
and $v_{n}^{1}\to v_1$ a.e. on $\mathbb{R}^N$.
Applying the compactness of the embedding
$H^1_0(B_1(0))\hookrightarrow L^2(B_1(0))$ and
$ \int_{B_1(0)}|v_{n}^{1}|^2>\frac{\delta}{2}$, we have
$\int_{B_1(0)}|v_{1}|^2\geq \frac{\delta}{2}$ and $v_1\neq 0$.
Because $u_{n}^{1}\rightharpoonup0$ a.e. on $H^1(\mathbb{R}^N)$,
so $(y_{n}^{1})_{n\geq1}$ must be unbounded.

Suppose $|y_{n}^{1}|\to \infty$ as $n\to \infty$. We claim
$I'_{\infty}(v_1)=0$ and $u_{n}^{2}:=u_{n}^{1}-v_{1}(\cdot-y_{n}^{1})$  such that
\begin{gather*}
\|u_{n}^{2}\|^2=\|u_n\|^2-\|v_0\|^2-\|v_1\|^2+o(1),\\
I_{\infty}(u_{n}^{2})\to C-I(v_0)-I_{\infty}(v_1),\\
I_{\infty}'(u_{n}^{2})\to 0 \ in\ H^{-1}(\mathbb{R}^N).
\end{gather*}
Indeed, since $v_{n}^{1}\rightharpoonup v_1$ weakly in $H^1(\mathbb{R}^N)$, we have
\[
\|u_{n}^{2}\|^2 = \|u_{n}^{1}(\cdot+y_{n}^{1})-v_1\|^2
= \|u_{n}^{1}\|^2-\|v_1\|^2+o(1).
\]
Applying Lemma \ref{minmax5}   and similar arguments as Step 2, we prove the claim.
In the sequel, we iterate the above procedure,
and then construct sequences $(v_j)$ and  $(y_{n}^{j})$ such that
$|y_{n}^{j}|\to \infty$ and $|y_{n}^{i}-y_{n}^{j}|\to\infty$ for $i\neq j$,
$n\to\infty$. Since $I(u_n)\to C$ and $I_{\infty}(v_j)\geq c_\infty$
for every nontrivial critical point $v_j$ of $I_\infty$, then the iteration
must terminate at some finite number of steps, which
completes the whole proof.
\end{proof}

\section{Non-radial case}

In this section, we  first  give some properties of the Nehari manifold
 $\mathcal{N} $ and the relationship between $c$ and $c_\infty$.
Some of similar results  can be found in \cite{vv} and \cite{sd}.
 Here we give the complete proof.

\begin{lemma}\label{lemm2.4}
Let  $N\geq3$, $\alpha\in(0,N)$ and $2\leq p<\frac{N+\alpha}{N-2}$.
If $q(x)$ satisfies (H1), then the following statements are true:
\begin{itemize}
\item[(i)] $\mathcal{ N}$ is nonempty. Moreover, for every
 $u\in H^1(\mathbb{R}^N)$ with $\mathbb{D}(u)>0$,
there exists a unique $t_u\in (0,\infty)$ such that $t_uu\in \mathcal{N} $ and
\begin{equation}
t_u=\Big(\frac{\|u\|^2}{\mathbb{D}(u)}\Big)^\frac{1}{2p-2}.
\end{equation}
Furthermore, $I(t_u u)=\sup_{t>0}I(tu)
 =(\frac{1}{2}-\frac{1}{2p})J^{\frac{p}{p-1}}(u)$.

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

\item[(iii)] $c>0$.

\item[(iv)] $\mathcal{N}$ is a $\mathcal{C}^2$-submanifold of $H^1(\mathbb{R}^N)$.

\item[(v)] $c\leq c_{\infty}$.
\end{itemize}
\end{lemma}

\begin{proof} (i) First, it follows from (H1)  that there exists $R>0$
large enough such that $q(x)>\frac{1}{2}q_{\infty}$ for $|x|>R$,
and then we can find $u\in H^1(\mathbb{R}^N)$ such that
$\mathbb{D}(u)>0$.
In addition, for $t>0$, we have
$$
\frac{d}{dt}I(tu)=\langle I'(tu),u\rangle
=t\|u\|^2-t^{2p-1}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{q(x)q(y)|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy.
$$
Then there exists a unique $t_u$ such that $\langle I'(t_uu),u\rangle=0$,
which yields that $t_uu\in\mathcal{N}$.
Since  the map $t\mapsto I(tu)$ is
increasing for $0<t<t_u$ and decreasing for $t>t_u$, we have
 $I(t_u u)=\sup_{t>0}I(tu)$.
Furthermore, it follows from direct calculation that
\begin{align*}
I(t_u u)
&= \frac{1}{2}\Big(\frac{\|u\|^2}{\mathbb{D}(u)}\Big)
^{\frac{2}{2p-2}}\|u\|^2- \frac{1}{2p}\Big(\frac{\|u\|^2}{\mathbb{D}(u)}
 \Big)^{\frac{2}{2p-2}}\\
&= \big(\frac{1}{2}-\frac{1}{2p}\big)\Big(\frac{\|u\|^2}{\mathbb{D}^{1/p}(u)}
\Big)^{\frac{p}{p-1}} \\
&= \big(\frac{1}{2}-\frac{1}{2p}\big)J^{\frac{p}{p-1}}(u).
\end{align*}

(ii)
Define $\mathcal{M}=\{u\in H^1(\mathbb{R}^N):\mathbb{D}(u)>0\}$.
For $u\in\mathcal{M}$, $t_u u\in \mathcal{N}$ and then
$I(t_u u)\geq\inf_{u\in \mathcal{N}}I(u)$, which concludes
$\inf_{u\in \mathcal{M}}\sup_{t>0} I(tu)\geq\inf_{u\in \mathcal{N}} I(u)$.
If $u\in {H^1(\mathbb{R}^N)\backslash \mathcal{M}}$ and $u\neq 0$, we have
$\sup_{t>0}I(tu)=\infty$. In the contrary,
if  $u\in\mathcal{N}$, then $t_u=1$ and hence
$I(u)\geq\inf_{u\in \mathcal{N}}\sup_{t>0}I(tu)$
which implies that $I(u)\geq\inf_{u\in H^1(\mathbb{R}^N)}\sup_{t>0}I(tu)$.
This yields a conclusion.

(iii) Let $\lambda >0$. For any $u\in\mathcal{N}$, it follows from (i)
above that $I(\frac{\lambda}{\|u\|}u)\leq I(u)$, and then
$I(u)\geq\inf_{v\in \mathcal{S}}I(v)$, where
$\mathcal{S}=\{v\in H^1(\mathbb{R}^N):\|v\|=\lambda\}$. Assume
$ \lambda=(\frac{p}{2C|q|^2_{L^\infty}})^\frac{1}{2p-2}$.
According to  \eqref{model-19} and
Sobolev embedding theorem, for any $v\in \mathcal{S}$, we have
\begin{equation}{\label{model-6}}
\begin{aligned}
I(v)
&\geq  \frac{1}{2}
\|v\|^2-\frac{C}{2p}|q|_{L^{\infty}}^2|v|_{L^{\frac{2Np}{N+\alpha}}}^p
 |v|_{L^{\frac{2Np}{N+\alpha}}}^p\\
&\geq  \frac{1}{2}
\|v\|^2-\frac{C}{2p}|q|_{L^{\infty}}^2\|v\|^{2p}\\
&= \lambda^2(\frac{1}{2}-\frac{\lambda^{2p-2}C|q|^2_{L^{\infty}}}{2p})
>0,
\end{aligned}
\end{equation}
where $C$ only depends on $p$, $N$ and $\alpha$.
This yields  $c>0$.

(iv) Denote $G: H^1(\mathbb{R}^N)\to \mathbb{R}$ by $ G(u)=\|u\|^2-\mathbb{D}(u)$.
Applying the same method as  Appendix B in \cite{sd}, we derive  $G$ is of
class $\mathcal{C}^2$ and its derivative is given by
$$
G'(u)v=2\langle u,v\rangle-2p\int_{\mathbb{R}^N}
\int_{\mathbb{R}^N}\frac{q(x)q(y)|u(y)|^p|u(x)|^{p-2}u(x)v(x)}{|x-y|^{N-\alpha}}
\,dx\,dy
$$
for all $u,\ v$ in $H^1(\mathbb{R}^N)$. Since $\mathcal{N}=G^{-1}(0)$ and
$$
G'(u)u=2\|u\|^2-2p\mathbb{D}(u)\neq0
$$
for all $u\in \mathcal{N}$, then we imply $0$ is a regular value of $G$.
This, combined with (ii) yields  that $\mathcal{N}$ is a submanifold of
class $\mathcal{C}^2$
of $H^1(\mathbb{R}^N)$ and $u\notin \ker G'(u)$ for $u\in \mathcal{N}$.



(v) By Lemma \ref{lemm2.1},  we assume  $w$ is a ground state solution
of \eqref{model-2} and $(x_n)_{n\geq1}$  is the unbounded sequence such
that $|x_n|\to\infty$, as $n\to \infty$. Then for every $w(\cdot-{x_n})$,
according to (i) above, there exists $t_n$ such that
$t_nw(\cdot-x_n)\in \mathcal{N}$ with
$$
t_n=\Big(\frac{\|w(x-x_n)\|^2}{\mathbb{D}(w(x-x_n))}\Big)^\frac{1}{2p-2}.
$$
Since $w\in \mathcal{N}_\infty$, by dominated convergence theorem, we have
$t_n\to 1$, and then
\begin{equation}{\label{model-14}}
\begin{aligned}
c&= \inf_{u\in \mathcal{N}} I(u)\leq I(t_n w(x-x_n))\\
&= \frac{1}{2} t_n^2
\|w(x-x_n)\|^2-\frac{1}{2p}|t_n|^{2p}\mathbb{D}(w(x-x_n))\\
&\to I_\infty(w)=c_\infty.
\end{aligned}
\end{equation}
This completes the proof.
\end{proof}

Next we show that the energy functional satisfies the Mountain-Pass geometry.

\begin{lemma}\label{minmax13}
Let  $N\geq3$, $\alpha\in(0,N)$ and $2\leq p<\frac{N+\alpha}{N-2}$.
Suppose $q$ satisfies {\rm (H1)}. Then the functional
$I$ satisfies the following conditions.
\begin{itemize}
\item[(i)] There exists $r>0$ such that $I(u)\geq \theta>0$ for all $\|u\|=r$.
\item[(ii)] There exists $e\in H^1(\mathbb{R}^N)$ such that $e\geq 0$,
$\|e\|>r$ and $I(e)<0$.
\end{itemize}
\end{lemma}

\begin{proof}
(i)By \eqref{model-19}, we have
\begin{equation}
\begin{aligned}
I(u)
&\geq \frac{1}{2}
\|u\|^2-\frac{C}{2p}|q|^2_{L^{\infty}}|u|_{L^{\frac{2Np}{N+\alpha}}}^p
 |u|_{L^{\frac{2Np}{N+\alpha}}}^p\\
&\geq  \frac{1}{2}
\|u\|^2-\frac{C}{2p}|q|^2_{L^{\infty}}\|u\|^{2p}.
\end{aligned}
\end{equation}
where $C$ only depends on $p$, $N$ and $\alpha$.
Since $p\geq 2$, we can choose $r,\theta>0$ such that $I(u)\geq \theta>0$
for all $\|u\|=r$.

(ii) Note that $I(0)=0$. In addition, we can find some $u\in H^1(\mathbb{R}^N)$
such that $\mathbb{D}(u)>0$. Then it follows from $p\geq 2$ that
$$
\lim_{t\to +\infty} I(tu)=\lim_{t\to +\infty}\left(\frac{1}{2}t^2\|u\|^2
-\frac{1}{2p}t^{2p}\mathbb{D}(u)\right)=-\infty.
$$
Hence, there exists $t_0>0$ such that $||t_0u||>r$ and $I(t_0u)<0$.
Take $e=|t_0u|$. Then the proof is complete.
\end{proof}

Define
 $$
c_1=\inf_{\gamma\in \Gamma}\max_{t\in[0,1]}I(\gamma(t)),
$$
 where $\Gamma=\{\gamma\in C([0,1],H^1(\mathbb{R}^N)):\gamma(0)=0,\gamma(1)=e\}$.
In  following lemma, we will show the relationship between $c$ and $c_1$
(see \cite[Proposition 2,14]{dingni}).

\begin{lemma}\label{minmax14}
Let  $N\geq3$, $\alpha\in(0,N)$ and $2\leq p<\frac{N+\alpha}{N-2}$.
 Suppose $q$ satisfies {\rm (H1)}. Then $c=c_1$.
\end{lemma}

\begin{proof}
According to Lemma \ref{minmax13}(i), there exists a small ball in
$H^1(\mathbb{R}^N)$ containing the origin such that
$I(u)\geq0$ for all $u$ in this component. By Lemma \ref{minmax13}(ii), we have
$$
\langle I'(e),e\rangle=2I(e)+(\frac{1}{p}-1)\mathbb{D}(e)<0.
$$
Thus  every $\gamma\in\Gamma$ has to cross $\mathcal{N}$ and $c\leq c_1$.

On the other hand, for any $\bar{u}\in \mathcal{N}$, let
 $l=\{t\bar{u}:t\geq0\}$ be a half-line
and $I(|\bar{u}|)=I(\bar{u})=\max_{u\in l}I(u)$ due to Lemma \ref{lemm2.4}(i).
Similarly, we denote $h=\{te:t\geq0\}$.
Let $V^+$ be the set $\{a|\bar{u}|+be:a\geq0,b\geq0\}$, let $V$ be the 2-dimensional
subsequence of $H^1(\mathbb{R}^N)$  spanned by $|\bar{u}|$ and $e$.
Note that $\mathbb{D}(|\bar{u}|)\neq0$ and $\mathbb{D}(e)\neq0$.
Then for any $v\in V^+\backslash\{0\}$, we have $\mathbb{D}(v)>0$.
Hence there exists a circle $S$ on $V$ with radius $R$ large enough such that
$I\leq 0$ on $S\bigcap V^+$.
 Suppose that $l$ and $h$ intersect $S$ at $v$ and $v_1$, respectively.
Thus we can find a path $\bar{\gamma}\in \Gamma$ through $v$ and $v_1$ such that
 $I(|\bar{u}|)=\max_{u\in \bar{\gamma}}I(u)$. Therefore, $c\geq c_1$.
This completes the proof.
\end{proof}

\begin{proposition}\label{minmax15}
Let  $N\geq3$, $\alpha\in(0,N)$ and $2\leq p<\frac{N+\alpha}{N-2}$.
Suppose $q$ satisfies {\rm (H1)}. If $c<c_{\infty}$ holds, then $I$ has a
critical point $u\in H^1(\mathbb{R}^N)$ such that $I(u)=c$.
\end{proposition}

\begin{proof}
By Lemma \ref{minmax13}, the energy functional  $I$ satisfies the mountain
 pass geometry. Due to Lemma \ref{minmax14} and the mountain pass theorem,
there exists a sequence $(u_n)_{n\geq1}\subset H^1(\mathbb{R}^N)$ such
that $I(u_n)\to c$ and $I'(u_n)\to 0$.
Since $c<c_\infty$, Lemma \ref{minmax16} completes the proof.
\end{proof}

Now, we are in a position to prove our main existence result.

\begin{proof}[Proof of Theorem \ref{minmax2}]
First we prove (i).  The  proof can be split into two cases.
\smallskip

\noindent\textbf{Case 1.} $q\equiv q_\infty$.
By scaling, the conclusion follows from Lemma \ref{lemm2.1}.
\smallskip

\noindent\textbf{Case 2.} $q\not\equiv q_\infty$. For any
$u\in H^1(\mathbb{R}^N)\backslash{\{0\}}$, it is easy to check
$I(u)<I_{\infty}(u)$. Due to  Lemma \ref{lemm2.1}, we have $c_\infty$
can be attained by a  $w\in H^1(\mathbb{R}^N)$.
By Lemma \ref{lemm2.4}(i),  there exists $t_w>0$ such that
$t_ww\in \mathcal{N}$.
Hence
$$
c\leq I(t_ww)<I_\infty(t_ww)\leq I_\infty(w)=c_\infty.
$$
Therefore, Proposition \ref{minmax15} yields our conclusion.

Next we  prove (ii). By way of contradiction, we assume $c$ can be
achieved by $\eta\in H^1(R^N)$.
 It follows from that Lemma \ref{lemm2.4}(i) that there exists
$t_\eta>0$ such that $t_\eta\eta\in \mathcal{N}_\infty$. Therefore,
$$
c=I(\eta)\geq I(t_\eta\eta)>I_\infty(t_\eta\eta)\geq c_\infty,
$$
a contradiction to Lemma \ref{lemm2.4}(v). This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{minmax25}]
 Set
\begin{equation}\label{minmax30}
\tilde{c}=\inf \{\|u\|^2:u \in H^1(\mathbb{R}^N),\; \mathbb{D}(u)=1\}.
\end{equation}
According to  the proof of Lemma \ref{lemm2.4}(i), it suffices to prove
 whether $\tilde{c}$ can be attained by some $u\in H^1(\mathbb{R}^N)$ or not.

First we prove (i). Without loss of generality, there exists a nonnegative
minimizing sequence $(u_n)_{n\geq1}$ such that $\|u_n\|^2\to \tilde{c}$
and $\mathbb{D}(u_n)=1$. Then going if necessary to a subsequence,
there exists $u_0\in H^1(\mathbb{R}^N)$ such that $u_n\rightharpoonup u_0$
weakly in $H^1(\mathbb{R}^N)$ and $u_n\to u_0$ a.e. on $\mathbb{R}^N$.
It is easy to see $u_0$ is nonnegative, $\|u_0\|^2\leq \tilde{c}$
and $\mathbb{D}(u_0)\leq 1$.

 Since $\lim_{|x|\to \infty}q(x)=0$, then for any $\varepsilon>0$,
we can find some  $R>0$  such that for any $|x|> R$, we have
$ |q(x)|<\varepsilon$. By Lemma \ref{minmax4}, we derive
$|u_n|^p\rightharpoonup |u_0|^p$
 weakly in $L^{\frac{2N}{N+\alpha}}(\mathbb{R}^N)$.
In addition,  $u_{n}\to u_0$ in $L_{\rm loc}^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$.
Then  by  Hardy-Littlewood-Sobolev inequality \eqref{model-19},  we have
\begin{equation}{\label{model-11}}
\begin{aligned}
&|\mathbb{D}(u_n)-\mathbb{D}(u_0)| \\
&\leq |\int_{\mathbb{R}^N\backslash B_R(0)}\int_{\mathbb{R}^N}
\frac{q(x)q(y)|u_{n}(y)|^p|u_{n}(x)|^p}{|x-y|^{N-\alpha}}\,dx\,dy\\
&\quad -\int_{\mathbb{R}^N\backslash B_R(0)}\int_{\mathbb{R}^N}
\frac{q(x)q(y)|u_0(y)|^p|u_0(x)|^p}{|x-y|^{N-\alpha}}\,dx\,dy|\\
&\quad +|\int_{B_R(0)}\int_{\mathbb{R}^N}
\frac{q(x)q(y)|u_n(y)|^p(|u_{n}(x)|^p-|u_0(x)|^p)}{|x-y|^{N-\alpha}}\,dx\,dy|\\
&\quad +|\int_{B_R(0)}\int_{\mathbb{R}^N}\frac{q(x)q(y)(|u_{n}
(y)|^p-|u_0(y)|^p)|u_0(x)|^p}{|x-y|^{N-\alpha}}\,dx\,dy|\\
&\to 0.
\end{aligned}
\end{equation}
Hence $\mathbb{D}(u_0)=1$. Let $u_{*}=\tilde{c}^{\frac{1}{2(p-1)}}u_0$.
Then according to the proof of Lemma \ref{lemm2.4}(i),
we conclude $I(u_{*})=c$ and $\langle I'(u_{*}),u_*\rangle=0$.
Therefore, the Lagrange multiplier rule yields that  $u_{*}$ is a ground
state  solution of \eqref{model-1}.

Now we prove (ii). Assume for contradiction,  $\tilde{c}$ is attained by
some $u\in H^1(\mathbb{R}^N)$ such that $\mathbb{D}(u)=1$ and
$\|u\|^2=\tilde{c}$.
Let $R>0$ and  $u_R(x)=u(x-x_R)$, where $x_R=(3R,0,\cdots,0)$. Clearly,
$\|u_R\|^2=\tilde{c}$, and it is easy to check that
\begin{equation}\label{minmax1}
\begin{aligned}
\mathbb{D}(u_R)
&\geq \int_{B_R(x_R)}\int_{B_R(x_R)}
 \frac{q(x)q(y)|u_R(x)|^p|u_R(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy\\
&\geq \min_{x\in B_R(x_R)}q(x)\min_{y\in B_R(x_R)}q(y)
 \int_{B_R(0)}\int_{B_R(0)}\frac{|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy\\
&\geq \min_{2R\leq|x|\leq 4R} q(x)\min_{2R\leq|y|\leq 4R} q(y)
 \int_{B_R(0)}\int_{B_R(0)}\frac{|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy\\
&\geq \max_{|x|\leq R} q(x)\max_{|y|\leq R} q(y)\int_{B_R(0)}
 \int_{B_R(0)}\frac{|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy\\
&= \int_{B_R(0)}\int_{B_R(0)}\frac{q(x)q(y)|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}
 \,dx\,dy+h(R).
\end{aligned}
\end{equation}
Here 
\[
h(R):= \int_{B_R(0)}\int_{B_R(0)}[\max_{|x|
 \leq R}q(x)\max_{|y|\leq R}q(y)-q(x)q(y)]
\frac{|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy\geq0.
\]
Since $q$ is not constant, we have
$h\not\equiv 0$ in $\mathbb{R}^N$. In addition, $h$ is nondecreasing in $R$
and bounded in $\mathbb{R}^N$ due to the fact that 
$q\in L^{\infty}(\mathbb{R}^N)$.
We assume $h(\infty)=\lim_{R\to\infty}h(R)$. 
Then there exist $R_1\geq R_0$ such that
$h(R)>\frac{1}{2}h(\infty)$ for all $R\geq R_1$. On the other hand,
since $\mathbb{D}(u)=1$, we can find  $R_2\geq R_0$ such that
 $$
\int_{B_R(0)}\int_{B_R(0)}
\frac{q(x)q(y)|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy>1-\frac{1}{2}h(\infty),
$$
for all $R\geq R_2$. Take $R=\max\{R_1, R_2\}$. Then $\mathbb{D}(u_R)>1$.
Let $\tilde{u}_R={(\frac{1}{\mathbb{D}(u_R)})}^{\frac{1}{2p}}u_R$.
Then $\mathbb{D}(\tilde{u}_R)=1$. But $\|\tilde{u}_R\|^2<\tilde{c}$,
 which contradicts  the
definition of $\tilde{c}$. This completes the proof.
\end{proof}

\section{Radial case}
It  is  known to us that if $q$ is radial and bounded in $\mathbb{R}^N$,
by standard  variational methods and the symmetric criticality principle
(see \cite[Theorem 1.28]{wm}),  we can find  a nontrivial radial solution
to \eqref{model-1} in $H^1(\mathbb{R}^N)$.  But in this section,
we consider the case that $q$ is radial and $q$ may be unbounded in $\mathbb{R}^N$.
 Applying a similar idea as that of \cite{dingni}, we obtain a nontrivial
nonnegative  radial solution for \eqref{model-1} in $H_r^1(\mathbb{R}^N)$.
First, we give the Radial lemma that will play a key role in the proof
of Theorem \ref{minmax21}. Throughout this section, we denote the norm of
$H^1_r(\mathbb{R}^N)$ ( or $H^1_{0,r}(B_k))$ by
$\|\cdot\|_{H^1_r(\mathbb{R}^N)}$ (or $\|\cdot\|_{H^1_{0,r}(B_k)}$, respectively).


\begin{lemma}[\cite{bhl}] \label{minmax17}
Let $N\geq2$. Then for any radial function $u\in H^1(\mathbb{R}^N)$,
$$
|u(r)|\leq C\|u\|r^{\frac{1-N}{2}},\quad \text{for}\ r\geq1,
$$
where $C$ only depends on $N$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{minmax21}]
First, we define
$$
M_{\infty}=\sup_{\|u\|_{H^1_r(\mathbb{R}^N)}=1}\mathbb{D}(u).
$$
It follows from  Lemma \ref{minmax17} and (H4) that $M_\infty< \infty$.
Without loss of generality, we assume there exists a nonnegative minimizing  sequence
$(u_n)_{n\geq 1}\subset H^1_r(\mathbb{R}^N)$ such that
$\|u_n\|_{H^1_r(\mathbb{R}^N)}=1$ and $\mathbb{D}(u_n)\to M_\infty$
as $n\to \infty$. Then going if necessary to a subsequence,
$u_n\rightharpoonup u_0$ weakly in $H^1(\mathbb{R}^N)$ and
$u_n\to u_0$ a.e. on $\mathbb{R}^N$. Obviously, $u_0$ is nonnegative, radial
and $\|u_0\|_{H^1_r(\mathbb{R}^N)}\leq 1$.
Applying Lemma \ref{minmax17} and (H4) again, we obtain
$(qu_n^p)_{n\geq1}$ is uniformly bounded in $L^{\frac{2N}{N+\alpha}}(\mathbb{R}^N)$,
and then  for any $\epsilon>0$, there exists $R>0$ such that
$|qu_n^p|_{L^{\frac{2N}{N+\alpha}}(\mathbb{R}^N\backslash {B_R(0)})}<\epsilon$.
Here $R$ is independent of $n$. Since $u_n\to u_0$ strongly in
$L_{\rm loc}^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$, by \eqref{model-19},
some standard argument yields that  $\mathbb{D}(u_n)\to \mathbb{D}(u_0)=M_\infty$.

Next we show $\|u_0\|_{H^1_r(\mathbb{R}^N)}=1$. If not, we can find
$\tilde{u}_0\in {H^1_{r}(\mathbb{R}^N)}$
such that $\|\tilde{u}_0\|_{H^1_{r}(\mathbb{R}^N)}=1$ and
$\tilde{u}_0=\lambda u_0$ with $\lambda>1$. This implies
$\mathbb{D}(\tilde{u}_0)>M_\infty$,
a contradiction to the definition of $M_\infty$.
Let $u_{*}=(\frac{1}{M_\infty})^{\frac{1}{2p-2}}u_0$.
According to  Lagrange multiplier rule, we conclude $u_*$
is a nonnegative radial  solution of \eqref{model-1} in $H^1_r(\mathbb{R}^N)$.
This completes  the proof.
\end{proof}

Theorem \ref{minmax26} can be treated as a by-product of Theorem \ref{minmax21}.
Now we give a simple proof. Let
$$\mathbb{D}_k(u)=\int_{B_k(0)}
\int_{B_k(0)}\frac{q(x)q(y)|u(x)|^p|u(y)|^p}{|x-y|^{N-\alpha}}\,dx\,dy.
$$

\begin{proof}[Proof of Theorem \ref{minmax26}]
This proof can be split into two steps.
\smallskip

\noindent\textbf{Step 1.}
Define
$$
M_{k,r}=\sup_{\|u\|_{H^1_{0,r}(B_k)}=1}\mathbb{D}_k(u), \quad
M_{k}=\sup_{\|u\|_{H^1_{0}(B_k)}=1}\mathbb{D}_k(u)
$$
Fix $k$. Without loss of generality, we assume there exists an nonnegative
minimizing sequence $(v_n^k)_{n\geq1}\subset H^1_{0,r}(B_k)$
such that $\|v_n^k\|_{H^1_{0,r}(B_k)=1}$ and
$\mathbb{D}_k(v_n^k)\to M_{k,r}$ as $n\to \infty$. Then going if necessary
to a subsequence, $v_n^k\rightharpoonup u_k$ weakly in $H^1_{0,r}(B_k)$ and
$v_n^k \to u_k$ a.e. on $\mathbb{R}^N$  as $n\to \infty$. Obviously,
$u_k$ is nonnegative, radial and $\|u_k\|_{H^1_{0,r}(B_k)}\leq 1$.
Since $v_n^k\to u_k$ strongly in $L_{\rm loc}^{\frac{2Np}{N+\alpha}}(\mathbb{R}^N)$,
by \eqref{model-19},
some standard arguments can imply that $\mathbb{D}_k(v_n^k)\to\mathbb{D}_k(u_k)$
as $n\to \infty$. Hence $\mathbb{D}_k(u_k)=M_{k,r}$. Similar to the proof of
Theorem \ref{minmax21},  we have $\|u_k\|_{H^1_{0,r}(B_k)}= 1$.
Therefore, $M_{k,r}$ is attained by $u_k\in H^1_{0,r}(B_k)$.
Let $w_k=\left(\frac{1}{M_{k,r}}\right)^{\frac{1}{2p-2}}u_k$.
It follows from Lagrange multiplier rule and symmetric criticality principle
(see \cite[Theorem 1.28]{wm}) that $w_k$ is a nontrivial nonnegative radial
solution of the  equation
\begin{equation}\label{minmax22}
\begin{gathered}
-\Delta u+u= q(x)(\int_{B_k(0)}
\frac{q(y)u^p(y)}{|x-y|^{N-\alpha}}dy)u^{p-1}\quad \text{in } B_k(0),\\
u\geq 0\quad  \text{in } B_k(0),\\
u=0\quad  \text{on } \partial{B_k}.
\end{gathered}
\end{equation}
 Similarly,  $M_k$ is also attained by
 $u_k^{*}\in H^1_{0}(B_k)$ and $ w_k^{*}=(\frac{1}{M_{k}})^{\frac{1}{2p-2}}u_k^{*}$.
\smallskip

\noindent\textbf{Step 2.}
Since $M_{k,r}$ is increasing with $k$, according to Theorem \ref{minmax21},
$$
\Big(\frac{1}{M_\infty}\Big)^{\frac{1}{2p-2}}\leq\|{w}_k\|_{H^1_{r}(\mathbb{R}^N)}
=\Big(\frac{1}{M_{k,r}}\Big)^{\frac{1}{2p-2}}
\leq \Big(\frac{1}{M_{1,r}}\Big)^{\frac{1}{2p-2}},
$$
that is to say, $({w}_k)_{k\geq1}$ is uniformly bounded in $H^1_{r}(\mathbb{R}^N)$.

On the other hand,  we choose $u_0\in H^1(\mathbb{R}^N)$ such that $u_0$
has compact support in $B_1(0)$ and $\|u_0\|=1$. Then for large $k$,
there exists $x_k\in \mathbb{R}^N$ such that $B_1(x_k)\subset B_k(0)$.
 Without loss of generality, we assume $|x_k|\to \infty$ as $k\to\infty$.  Hence
\begin{equation}
\begin{aligned}
M_k &\geq \int_{B_k(0)}\int_{B_k(0)}
\frac{q(x)q(y)|u_0(y-x_k)|^p|u_0(x-x_k)|^{p}}{|x-y|^{N-\alpha}}\,dx\,dy\\
&\geq \int_{B_1(x_k)}\int_{B_1(x_k)}
\frac{q(x)q(y)|u_0(y-x_k)|^p|u_0(x-x_k)|^{p}}{|x-y|^{N-\alpha}}\,dx\,dy\\
&\geq \min_{x\in B_1(x_k)}q(x)\min_{y\in B_1(x_k)}q(y)
\int_{B_1(0)}\int_{B_1(0)}
\frac{|u_0(y)|^p|u_0(x)|^{p}}{|x-y|^{N-\alpha}}\,dx\,dy.
\end{aligned}
\end{equation}
This implies that $M_k\to \infty$ as $k\to\infty$.
Then for $k$ large enough, \eqref{minmax22} has two different
 weak solutions $w_k$ and $w_k^*$. One is radial and the other is not.
This completes the proof.
\end{proof}

\subsection*{Acknowledgments}
This research was supported  by the
National Natural Science Foundation of  China (Grants 
11171098 and 11571371),
 and by the Hunan Provincial NSF (Grant No. 11JJ1001)

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\end{document}