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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 339, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/339\hfil Schr\"odinger-Kirchhoff type equations]
{Infinitely many  solutions for Schr\"odinger-Kirchhoff type equations
 involving the fractional $p$-Laplacian and critical exponent}

\author[L. Wang, B. Zhang \hfil EJDE-2016/339\hfilneg]
{Li Wang, Binlin Zhang}

\address{Li Wang \newline
School of Basic Science,
East China Jiaotong University,
Nanchang 330013,  China}
\email{wangli.423@163.com}

\address{Binlin Zhang (corresponding author)\newline
Department of Mathematics,
Heilongjiang Institute of Technology,
Harbin 150050,  China}
\email{zhangbinlin2012@163.com}

\thanks{Submitted October 11, 2016. Published December 30, 2016.}
\subjclass[2010]{35R11, 35A15, 47G20}
\keywords{Schr\"odinger-Kirchhoff type equation; fractional $p$-Laplacian; 
\hfill\break\indent critical Sobolev exponent}

\begin{abstract}
 In this article, we show the existence of infinitely many solutions for the
 fractional $p$-Laplacian equations of Schr\"odinger-Kirchhoff type equation
 $$
 M([u]_{s, p}^p) (-\Delta )_p^s  u+V(x)|u|^{p-2}u=
 \alpha |u|^{ p_s^{*}-2 }u+\beta k(x)|u|^{q-2}u \quad  x\in \mathbb{R}^N,
 $$
 where $(-\Delta )^s_p$ is the fractional $p$-Laplacian operator, $[u]_{s,p}$
 is the Gagliardo $p$-seminorm, $0 < s< 1<p<\infty$, $N> sp$,
 $ 1<q<p$, $M$ is a continuous and positive function, $V$ is a continuous
 and positive potential function and  $k(x)$ is a non-negative function in
 an appropriate Lebesgue space. By means of the concentration-compactness
 principle in fractional Sobolev space and Kajikiya's new version of the
 symmetric mountain pass lemma, we obtain the existence of infinitely many
 solutions which tend to zero for suitable positive parameters $\alpha$ and
 $\beta$.
\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 and statement of main result}

In this article, we consider the following fractional $p$-Laplacian equations of
Schr\"odinger-Kirchhoff type:
\begin{equation} \label{eqS1.1}
\begin{gathered}
M([u]_{s,p}^p)(-\Delta)^s_pu+V(x)|u|^{p-2}u
=\alpha |u|^{ p_s^{*}-2 }u+\beta k(x)|u|^{q-2}u\quad \text{in }\mathbb{R}^N,\\
[u]_{s,p}^p:=\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dx\,dy,
\end{gathered}
\end{equation}
where $0 < s< 1<p<\infty$, $1<q<p$, $N> sp$, $p_s^{*}=\frac{Np}{N-ps}$
is the fractional critical Sobolev exponent, $M, V$ and  $k$ are functions
satisfying some suitable conditions which will be given later,
$ (-  \Delta )_p^s$ is the fractional $p$-Laplace operator which, up to
normalization factors, by the Riesz potential as
$$
(-  \Delta )_p^s u(x) :=
2 \lim_{\epsilon\to 0} \int_{\mathbb{R}^N\backslash B_{\epsilon}(x)}
\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N
 +ps}}\mathrm{d}y,\quad x\in \mathbb{R}^N,
$$
where
 $B_\epsilon(x) := \{y \in \mathbb{R}^N : |x - y| < \epsilon\}$.
Consistent, up to some normalization constant depending upon $n$ and $s$,
with the linear fractional Laplacian $(-\Delta )^s$ in the case $p = 2$.
As for some recent results on the fractional
$p$-Laplacian, we refer to for example \cite{ASM, XZR, XZR2} and the
references therein.

Recently, a great deal of attention has been focused on studying of problems
involving fractional Sobolev spaces and corresponding nonlocal equations,
both from a pure mathematical point of view and for
concrete applications, since they naturally arise in many different contexts,
 such as, among the others, the thin obstacle problem, optimization, finance,
phase transitions, stratified materials, anomalous diffusion,
crystal dislocation, soft thin films, semipermeable membranes, flame propagation,
conservation laws, ultra relativistic limits of quantum mechanics,
quasi-geostrophic flows, multiple scattering, minimal surfaces,
materials science and water waves. For more details, we can see
\cite{LAC, EGE, MRS} and the references therein.

Problem \eqref{eqS1.1} is related to the stationary analogue of the  Kirchhoff model
\begin{align*}
\rho\frac{\partial ^2u}{\partial t^2}
-\Big(\frac{p_0}{\lambda}+\frac{E}{2L}\int_0^L|\frac{\partial u}{\partial x}|^2dx
\Big)\frac{\partial ^2u}{\partial x^2}=0
\end{align*}
which was proposed by  Kirchhoff in 1883 as a generalization of the
well-known D'Alembert wave equation
for free vibrations of elastic strings, where $\rho$, $p_0$, $\lambda$, $E$, $L$
are constants which represent some physical
meanings respectively. Indeed, Kirchhoff's model takes into account the changes
in length of the string produced by transverse vibrations.
In particular, Kirchhoff's equation models several physical and biological systems,
we refer to \cite{ACM} for more details.
Recently,  Fiscella and Valdinoci \cite{FAVE} proposed a stationary
Kirchhoff model involving the fractional Laplacian
by taking into account the nonlocal aspect of the tension arising
from nonlocal measurements of the fractional length of the string,
see \cite[Appendix A]{FAVE} for further details.

When $p=2$ and $M\equiv1$, problem \eqref{eqS1.1} becomes the fractional
Schr\"odinger equation with a critical nonlinearity
\begin{align}\label{eq11}
(-\Delta)^{s} u+V(x)u=\alpha |u|^{ p_s^{*}-2 }u+\beta k(x)|u|^{q-2}u\quad  \text{in }
 \mathbb{R}^N,
\end{align}
which was first proposed by  Laskin in \cite{Laskin1, Laskin2} as a result of
expanding the Feynman path integral, from the Brownian-like to the L\'{e}vy-like
quantum mechanical paths. In recent years, a lot of interesting results
about problem \eqref{eq11} have been obtained, here we just quote a few,
see for example \cite{MR3, ZZX, ZZR}.

For our problem, we first assume that the Kirchhoff function
$M : \mathbb{R}^+_0 \to \mathbb{R}^+$, the potential function $V(x)$ and the
weight function $k(x)$ satisfy the following assumptions:
\begin{itemize}
\item[(A1)] $M\in C( \mathbb{R}^+_0,\mathbb{R}^+) $ satisfies
$\inf_{t\in \mathbb{R}^+_0} M(t) \geq m_0 > 0$,  where  $m_0$  is a constant.

\item[(A2)]  There exists   $\theta \in [1,\frac{N}{N-ps} )$ such that
$ \theta \mathcal{M}(t) := \theta \int_{0}^t  M(\tau )\mathrm{d}\tau \geq M(t)t$
 for any  $t \in\mathbb{R}^+_0$.

\item[(A3)]  $V \in  C(\mathbb{R}^N )$ satisfies
$\inf_{ x\in \mathbb{R}^N} V(x) \geq V_0 > 0$, where $V_0 > 0$ is a constant.

\item[(A4)] $0\leq k(x)\in   L^{r}(\mathbb{R}^N)$, where
$r=\frac{p_s^{*}}{p_s^{*}-q}$.

\end{itemize}
A typical example for $M$ is  $M(t) = m_0 + b_1t^{\theta-1}$ with $\theta\ge 1$,
$ m_0 \in \mathbb{R}^+$ and $b_1 \in \mathbb{R}^+_0$.
When $M$ is of this type, the Kirchhoff problem
is said to be  \emph{non-degenerate} if $m_0>0$, while it is
called \emph{degenerate} if $m_0=0$.

Next we state some recent advance related with our problem.
First of all, we consider the case that $M$ satisfies (A1) and (A2).
 Xiang, Zhang and Ferrara \cite{XZF}
studied the existence of solutions for the following Kirchhoff type problem
driven by the fractional $p$-Laplacian operator  with homogeneous Dirichlet
boundary conditions:
\begin{equation} \label{eq1}
\begin{gathered}
M([u]_{s, p}^p)(-\Delta )_p^su= f(x, u)\quad \text{in }\Omega,\\
u=0\quad\text{in } \mathbb{R}^{N}\setminus\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is an open bounded subset of $\mathbb{R}^{N}$ with smooth
boundary $\partial\Omega$. By using variational methods,
they gave some existence results with respect to $f(x,u)=a(x)|u|^{q-2}u$
with $1<q<p$ and $p<q<p_s^*$.
In \cite{XZG},  Xiang,  Zhang and  Guo obtained the existence
of infinitely many solutions for problem
\eqref{eq1} with $p=2$ by applying the fountain theorem and the dual fountain
theorem. More precisely, they considered mainly two cases:
for any $\lambda \in \mathbb{R}$, the above result holds as
$f(x,u)=|u|^{q-2}u+\lambda u$ with $q\in (2, 2^*_s)$;
there exists $\Lambda^{*}>0$ such that for any for $ \lambda \in (0, \Lambda^{*})$,
the above result holds
when $f(x,u)=\alpha|u|^{\xi-2}u+\beta |u|^{\eta-2}u+\lambda u$ for any
$\alpha\in \mathbb{R}, \beta>0$ or
for any $\alpha>0, \beta\in \mathbb{R}$, where $1<\xi<2\leq 2\theta<\eta<2^{*}_{s}$,
see also \cite{BMS} for similar applications of the fountain theorem.
By appealing to Krasnoselskii's genus theory,  Fiscella in \cite{FA}
 obtained the existence of infinitely many solutions for problem \eqref{eq1}
with $p=2$ and $f(x,u)=\lambda g(x,u) [\int_{\Omega}G(x, u)dx]^r+|u|^{2^*_s-2}u$,
where $G(x, u)=\int_{0}^{u}g(x,\mu)d\mu$, $r$ and $\lambda$ are positive parameters,
see also \cite{FMS, MoRe, MMTZ, PXZ2} for similar results involving variational
methods.
By using a truncation argument and the mountain pass theorem,  Autuori,  Fiscella
 and  Pucci \cite{AFP} considered the existence of solutions for problem
\eqref{eq1} with $f(x,u)=\lambda g(x,u)+ |u|^{2^*_s-2}u$ in the degenerate
and non-degenerate cases.

On the other hand,  Pucci,  Xiang and  Zhang \cite{PXZ} were concerned with
 the nonhomogeneous Schr\"odinger-Kirchhoff type equations involving the
fractional $p$-Laplacian
\begin{align}\label{k1}
M([u]_{s,p}^p)(-\Delta)^s_pu&+V(x)|u|^{p-2}u=f(x,u)+g(x)\quad \text{in }\mathbb{R}^N,
\end{align}
where $M$ satisfies (A1) and (A2), $f(x,u)$ satisfied the subcritical
growth. with the help of the Ekeland variational principle and the mountain
pass theorem, the authors obtained the existence of at least two solutions
for problem \eqref{k1}, see also \cite{BM} for related results.
Subsequently,  in \cite{PXZ2} they considered the existence and multiplicity
 of solutions for the  equation
\begin{equation}\label{k2}
M([u]_{s,p}^p)(-\Delta)^s_pu+V(x)|u|^{p-2}u
=\lambda \omega(x)|u|^{q-2}u-h(x)|u|^{r-2}u\quad \text{in }\mathbb{R}^N,
\end{equation}
where $h(x)$ is a non-negative function satisfying some ratio of integration
with $\omega(x)$, $1<q<r<\infty$, see also \cite{PS, XZR} for related results.
In this case, the existence of infinitely many solutions for problem \eqref{k2}
 was obtained by genus theory in the degenerate case.

In \cite{S1}, with the help of classical variational techniques,  Servadei
 consider the existence of infinite solutions for problem \eqref{eq1},
in which $M\equiv 1$ and $f(x,t)=|u|^{q-2}u$ with $2<q<(2N-2s)/(N-2s)$,
but in presence of a perturbation $h\in L^2(\Omega)$.
In \cite{MB1}, by means of the symmetric mountain pass theorem,
 Molica Bisci obtained the existence of infinite solutions for\eqref{eq1}
with $M\equiv 1$. Concerning the study of elliptic equations with critical
 Sobolev exponent, we refer to the seminal works of  Br\'{e}zis and
Nirenberg in \cite{NB}. In order to overcome the lack of compactness,
 Lions in \cite{PL1, PL2} developed the concentrate-compactness principle.
 Based on the principle of concentration compactness in the fractional
 Sobolev space in \cite{PP},  Zhang,  Zhang and  Xiang \cite{ZZX} obtained
the existence of ground state solution for problem \eqref{eq11} with $\alpha=1$,
see also \cite{ZZR} for extensive discussions for this kind of problem.
  Xiang, Zhang and  Zhang \cite{XZZ} studied the multiplicity of solutions
for problem \eqref{eqS1.1} in some special cases.
For this, they extended the concentrate-compactness principle in \cite{PP}
to the setting of fractional $p$-Laplacian.
In the context of fractional Laplacian, the discussions about the existence
 of infinitely many solutions, we also refer to \cite{FMS2}.


Motivated by the above works, in the present paper we are
interested in the existence of infinitely many solutions  for problem \eqref{eqS1.1}
 by means of Kajikiya's new version of the symmetric mountain pass lemma.
To our best knowledge, there is no result in the literature on problem \eqref{eqS1.1}.
There is no doubt that we encounter serious difficulties because of the lack of
compactness and of the nonlocal nature of the fractional $p$-Laplacian.
To this end, we will use the concentrate-compactness principle in \cite{XZZ} to conquer the difficulty due to the lack of
compactness.

 Now we first give the definition of weak solutions for
problem \eqref{eqS1.1}.

\begin{definition}\label{d1.1}\rm
  We say that $u\in W$ is a weak solution of problem \eqref{eqS1.1}, if $u\in W$
 and
  \begin{align*}
& M([u]_{s,p}^p)\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))
 (\varphi(x)-\varphi(y))}{|x-y|^{N +ps}}\mathrm{d}x\mathrm{d}y\\
&+ \int_{\mathbb{R}^N}V(x)|u|^{p-2}u\varphi\mathrm{d}x \\
&=\alpha\int_{\mathbb{R}^N}|u|^{p_s^{*}-2}u\varphi\mathrm{d}x
 +\beta\int_{\mathbb{R}^N} k(x)|u|^{q-2}u\varphi \mathrm{d}x
 \end{align*}
for all $\varphi \in  W$.
\end{definition}

In the sequel we will omit the term weak when referring to solutions
that satisfy the conditions of   Definition \ref{d1.1}.
 Our  main result  of this paper is stated as follows.

\begin{theorem}\label{T1.1}
  Let  {\rm (A1)--(A4)} and  $1 <q< p$ hold. Then
\begin{itemize}
\item[(i)] for all $\alpha> 0$ there exists $\beta_0 > 0$ such that if
 $0 < \beta < \beta_0$, then   \eqref{eqS1.1} has a sequence
of solutions $\{u_n\}_n$ with $I(u_n) < 0$,
$I(u_n) \to 0$ and $\lim_{n\to \infty} u_n\to 0$.

\item[(ii)] for all $\beta > 0$ there exists  $\alpha_0 > 0$ such that if
$0 <\alpha<\alpha_0$, then \eqref{eqS1.1} has a sequence
of  solutions $\{u_n\}_n$ with $I(u_n) < 0$, $I(u_n) \to 0$ and
 $\lim_{n\to \infty} u_n\to 0$.
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
From Theorem \ref{T1.1} it is natural to raise the following open problems:
(i) What if $\theta p<q<p^*_s$?
(ii) Are our result still valid in the degenerate case?
These problems would be investigated by the authors in future works.
\end{remark}

 The rest of this paper is organized as follows.
The functional framework and some preliminaries are given in Section  2.
 In Section 3, behavior of $(PS)$ sequences are established.
 The proof of  the main result Theorem
\ref{T1.1} is given in Section 4.

 $L^{t}(\mathbb{R}^N)$ is the usual Lebesgue space  with the norm
  $\|u\|_p^{p}=\int_{\mathbb{R}^N}|u|^{p} \mathrm{d}x, 1\leq p<+\infty$.
 Various positive constants are denoted by $C$ and $ C_{i}$.

\section{Preliminaries}

In this section, we first give some basic results of fractional Sobolev space
and then provide some useful technical lemmas, which will be used
in the sequel.

Let $0 < s < 1 < p < \infty$ be real numbers.
The Gagliardo seminorm is defined for all measurable function
$u : \mathbb{R}^N \to  \mathbb{R}$ by
\begin{equation}\label{eqS3.1.'}
[u]_{s,p} =\Big(
\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y\Big)^{1/p}.
\end{equation}
The fractional Sobolev space is defined as
$$
W^{s,p} (\mathbb{R}^N)
=\big\{u\in L^p(\mathbb{R}^N) :
\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y<\infty
 \big\},
$$
equipped with the norm
 \begin{equation}\label{eqS1.2}
 \|u\|_{W^{s,p} (\mathbb{R}^N)}= \Big(
\|u\|^p_{p}+ [u]_{s,p}^p \Big)^{1/p}.
 \end{equation}
 As it is well-known that this space is a uniformly convex Banach
space. For a detailed account on the properties of $W^{s,p} (\mathbb{R}^N)$,
we refer to \cite{EGE}.

Let $ W$ denote the completion of $C_0^{\infty} (\mathbb{R}^N)$, with respect
to the norm
 \begin{equation}\label{eqS1.3}
 \|u\|_{W} : =\Big([u]_{s,p}^p+\|u\|^p_{p,V}  \Big)^{1/p},\quad
\|u\|^p_{p,V}  =\int_{\mathbb{R}^N}V(x)|u|^{p} \mathrm{d}x.
 \end{equation}
Clearly the definition makes sense since every
$\varphi\in C_0^{\infty}(\mathbb{R}^N)$ has finite Gagliardo norm as
well finite norm $\|\varphi\|_{p,V} $. Indeed,
$L^p(\mathbb{R}^N , V) = (L^p(\mathbb{R}^N , V), \|\cdot\|_{p,V}  )$
is a uniformly convex
Banach space thanks to $(V1)$. By standard arguments, it is clear that
$W$ is a uniformly convex
Banach space, see  \cite[Lemma 10]{PXZ} for details.
The embedding $W  \hookrightarrow  L^t(\mathbb{R}^N)$ is continuous
for any $t\in   [p,p_s^{*}]$ by \cite[Theorem 6.7]{EGE};
that is, there exists a positive constant $ C_*$ such that
 \begin{equation}\label{eqS1.4}
 \|u\|_{L^t(\mathbb{R}^N)}\le C_*\|u\|_{W} \quad \text{for all } u \in W.
 \end{equation}

In our context, the Sobolev constant is given by
\begin{equation}\label{eqS2.,4}
S:= \inf_{u\in D^{s,p}(\mathbb{R}^N)\setminus \{0\}}
\frac{\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}} {|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y}{(\int_{\mathbb{R}^N}
|u|^{p_s^{*}}\mathrm{d}x)^{p/p_s^*}}
 \end{equation}
is the associated Rayleigh quotient.
The constant $S$ is well defined, as can be seen in \cite[Theorem 7.58]{Ra}.

Next we recall the the concentration-compactness
principle in the setting of the fractional $p$-Laplacian, see
 \cite[Definition 2.1,Theorem 2.1 and Theorem 2.2]{XZZ}.

 \begin{definition} \label{d1.2}  \rm
Let $\mathcal{\widetilde{M}}(\mathbb{R} )$ denote the finite nonnegative Borel
measure space on $\mathbb{R}^N $. For any
$\mu \in \mathcal{\widetilde{M}}(\mathbb{R}^N), \mu(\mathbb{R}^N) = \|\mu \|$ holds.
We say that $\mu \rightharpoonup \mu$ weakly $\ast$ in
$\mathcal{\widetilde{M}}(\mathbb{R}^N)$, if $(\mu_n, \eta) \to  (\mu, \eta)$ holds
for all $\eta \in C_0(\mathbb{R}^N)$ as $n \to \infty$.
 \end{definition}

 \begin{proposition} \label{p1.1}
Let $\{u_n\}_n  \subset D^{s,p}(\mathbb{R}^N)$ with upper bound $C > 0$
for all $n \geq 1$ and
\begin{gather*}
u_n \rightharpoonup u \quad  \text{weakly in }  D^{s,p}(\mathbb{R}^N),\\
\int_{\mathbb{R}^{N}}\frac{|u_n(x)-u_n(y)|^{p}} {|x-y|^{N
 +ps}}\mathrm{d}y\rightharpoonup \mu  \quad \text{weak $\ast$ in }
 \mathcal{\widetilde{M}}(\mathbb{R}^N),\\
|u_n(x)|^{p^*_s}\rightharpoonup \nu \quad  \text{weak $\ast$ in }
 \mathcal{\widetilde{M}}(\mathbb{R}^N).
\end{gather*}
Then
\begin{gather*}
\mu =\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}} {|x-y|^{N
 +ps}}\mathrm{d}y+\sum_{ j\in \mathcal{J}}\mu_j \delta_{x_j}
+\tilde{\mu},\quad \mu(\mathbb{R}^N) \leq  C^p,
\\
\nu = |u|^{p^*_s}+\sum_{ j\in \mathcal{J}}\nu_j \delta_{x_j},\quad
 \nu(\mathbb{R}^N) \leq S^{p^*_s} C^p,
\end{gather*}
where $J$ is at most countable, sequences
$\{\mu_j\}_j , \{\nu_j\}_j  \subset \mathbb{R}^+_0,
 \{x_j\}_j \subset \mathbb{R}^N,\delta_{x_j}$ is the Dirac mass centered
at $ x_j$,   $\tilde{\mu}$ is a non-atomic measure,
\begin{gather*}
\nu(\mathbb{R}^N) \leq S^{-\frac{p_{s}^*}{p}}\mu(\mathbb{R}^N)^{\frac{p_{s}^*}{p}},\\
\nu_j \leq S^{-\frac{p^*_s}{p}}\mu_j^{\frac{p_{s}^*}{p}},\quad \forall j \in J,
\end{gather*}
and $S > 0$ is the best constant of
$D^{s,p}(\mathbb{R}^N)\hookrightarrow  L^{p^*_s}(\mathbb{R}^N) $
 \end{proposition}

 \begin{proposition} \label{p1.2}
  Let $\{u_n\}_n  \subset D^{s,p}(\mathbb{R}^N)$  be a bounded sequence such that
\begin{gather*}
\int_{\mathbb{R}^{N}}\frac{|u_n(x)-u_n(y)|^{p}} {|x-y|^{N
 +ps}}\mathrm{d}y\rightharpoonup \mu \quad  \text{weak $\ast$ in }
    \mathcal{\widetilde{M}}(\mathbb{R}^N), \\
|u_n(x)|^{p^*_s}\rightharpoonup \nu \quad  \text{weak $\ast$ in }
    \mathcal{\widetilde{M}}(\mathbb{R}^N),
\end{gather*}
 and define
 \begin{gather*}
\mu_{\infty}:   =\lim_{R\to \infty} \limsup_{n\to \infty}
   \int_{\{x\in \mathbb{R}^N:|x|>R\}}
\int_{\mathbb{R}^N}\frac{ |u_n(x)-u_n(y)| ^{p}}{|x-y|^{N+ps}}\mathrm{d}y\mathrm{d}x,
\\
\nu_{\infty} :=\lim_{R\to \infty} \limsup_{n\to \infty}
\int_{\{x\in \mathbb{R}^N:|x|>R\}} |u_n| ^{p^*_s} \mathrm{d}x.
    \end{gather*}
Then the quantities $\mu_{\infty} $ and $\nu_{\infty} $ are well defined and satisfy
  \begin{gather*}
   \limsup_{n\to \infty} \iint_{\mathbb{R}^{2N}}\frac{ |u_n(x)-u_n(y)| ^{p}}
 {|x-y|^{N+ps}}\mathrm{d}y\mathrm{d}x
 = \int_{\mathbb{R}^N}\mathrm{d}\mu+ \mu_{\infty},    \\
 \limsup_{n\to \infty} \int_{ \mathbb{R}^N}|u_n| ^{p^*_s} \mathrm{d}x
=   \int_{\mathbb{R}^N}\mathrm{d}\nu+ \nu_{\infty}.
\end{gather*}
Moreover,
$$
S\nu_{\infty}^{\frac{p}{p^*_s}}\leq\mu_{\infty}.
$$
 \end{proposition}

\begin{lemma}[{\cite[Lemma 2.3]{XZZ}}] \label{lA.1}
Assume $\{u_n\}_n \subset D^{s,p}(\mathbb{R}^N)$ is the sequence given
by Lemma \ref{l2.5} and   for $\varepsilon> 0$, let
 $\phi_j(x)$ be a smooth cut-off function
 centered at $x_j$ such that  $ 0\le \phi_j(x)\le 1$,
$\phi_j(x)\equiv 0$ on  $|x-x_j|\ge 2$, $\phi_j (x)\equiv 1$ on
$|x-x_j|\le 1 $, and $|\nabla \phi_j(x)|\le  2$  for all
  $x\in \mathbb{R}^N$. Set
 $\phi_j^{\varepsilon}(x)=\phi_j(x/\varepsilon)$ for all $x\in \mathbb{R}^N$. Then
 $$
\lim_{\varepsilon\to 0}\limsup_{n\to \infty}
\iint_{\mathbb{R}^{2N}}\frac{|\phi_j^{\varepsilon}(x)-\phi_j^{\varepsilon}(y)|^p
|u_n(y)|^p} {|x-y|^{N +ps}}\mathrm{d}x\mathrm{d}y =0.
$$
\end{lemma}

\begin{lemma}[{\cite[Lemma\ 2]{PXZ}}] \label{Lemma2}
 Let $(\mathbf{M1})$  and   $(\mathbf{V1})$  hold. Then $J : W \to \mathbb{R}$
is  of class $C^1(W)$ and
\begin{align*}
 \langle J'(u), v\rangle
& =  M([u]_{s,p}^p) \iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p-2 }
(u(x)-u(y))(v(x)-v(y)) }{|x-y|^{N +ps}}\mathrm{d}x\mathrm{d}y\\
 &\quad + \int_{\mathbb{R}^N}V(x)|u(x)|^{p-2} u(x)v(x)\mathrm{d}x ,
 \end{align*}
for all $u, v \in W$. Moreover, $J$ is weakly lower semi-continuous in $ W$.
\end{lemma}

\begin{lemma}[{\cite[Lemma\ 2.1]{WWK}}] \label{l2.4}
For each $k$ in $L^r(\mathbb{R}^N)$, the functional
$\mathcal{F}(u) =  \int_{\mathbb{R}^N} k(x)|u|^q\mathrm{d}x$ is well
defined and weakly continuous on $W$. Moreover, $\mathcal{F}(u)$
is continuously differentiable, its derivative $\mathcal{F}' : W \to W^* $
is given by
$$
\langle \mathcal{F}'(u), \varphi \rangle
= q \int_{\mathbb{R}^N}k(x)|u|^{q-2}  u \varphi \mathrm{d}x,\quad
\forall  \varphi \in W .
$$
 \end{lemma}

 \begin{lemma}[{\cite[Theorem2.1]{PXZ}}] \label{lemma4.6}
 Let  {\rm (V1)} hold. Let $\vartheta\in  [p,p^{*}_s)$ be a fixed
exponent and let $\{v_j \}_j$ be a bounded sequence in $W$.
Then there exists $v \in W\cap L^{\vartheta}(\mathbb{R}^N )$ such that up
to a subsequence, $v_j \to v$ strongly in $L^{\vartheta}(\mathbb{R}^N )$ as
 $j\to \infty$.
\end{lemma}

\section{Behavior of (PS) sequences}

In this  section, we perform a careful analysis of the behavior of minimizing
sequences with the aid of the concentration--compactness principle in
fractional Sobolev space stated above, which allows to recover
compactness below some critical threshold.

Let $E$ be a real Banach space and $I: E\to \mathbb{R}$ be a function of
 class $C^1$. We say that $\{u_n\}_n\subset E$
is a $(PS)_c$ sequence if $I(u_n) \to c$ and $I'(u_n)\to 0$. $I$
 is said to satisfy  the  Palais-Smale
condition at level $c$ ($(PS)_c$ in short) if
any $(PS)_c$ sequence contains a convergent subsequence.

\begin{lemma}\label{l2.5}
  Let  {\rm (A1)--(A4)}, $1<q\leq p $ and $ c < 0$ hold.
Then
\begin{itemize}
\item[(i)] there exists $C > 0$ such that, for all $n\in \mathbb{N}$, $\|u_n
 \|_{ W }\le C$;

\item[(ii)]  for each $\alpha> 0$ there exists $\beta_* > 0$ such that if $0 <  \beta
< \beta _*$, then $I$ satisfies $(PS)_c$;

\item[(iii)]   for each $\beta > 0$ there exists  $\alpha_* > 0$ such that if
$0 <\alpha<\alpha_*$, then $I$ satisfies $(PS)_c$.
\end{itemize}
 \end{lemma}

\begin{proof}
 We first prove that $\{u_n\}_n$ is bounded in   $W$.
Let $\{u_n\}_n$ be a $(PS)_c$ sequence in $ W $ such that for all
$ \varphi \in C^{\infty}_{0}(\mathbb{R}^N)$,
\begin{gather}\label{eqS2.3}
\begin{aligned}
&c +o_n(\| u_n\|_{W})=I(u_n) \\
& = \frac{1}{p}\Big[ \mathcal{M}([u_n]_{s,p}^p)+ \|u_n\|_{p,V}^p\Big]
-\frac{\alpha}{p^{*}_s}\int_{\mathbb{R}^N} |u_n|^{p^{*}_s}\mathrm{d}x
-\frac{\beta}{q}\int_{\mathbb{R}^N} k(x)|u_n|^q\mathrm{d}x .
   \end{aligned} \\
\label{eqS2.4}
\begin{aligned}
 o_n(\|u_n \|_{W})
&= \langle I'(u_n), u_n \rangle\\
&= M([u_n]_{s,p}^p) \iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u_n(y)|^{p } }{|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y
+ \int_{\mathbb{R}^N}V(x)|u_n|^{p} \mathrm{d}x  \\
&\quad-\alpha\int_{\mathbb{R}^N}|u_n|^{p_s^{*} }  \mathrm{d}x
-\beta\int_{\mathbb{R}^N} k(x)|u_n|^{q }  \mathrm{d}x.
   \end{aligned}
\end{gather}
 Therefore,
% \label{eqS2.6}
\begin{align*}
 0 &>c +o_n( \|u_n\|_W)
   =  I(u_n)-\frac{1}{p_s^{*}} \langle I'(u_n), u_n \rangle\\
&=  \frac{1}{p  }\mathcal{M}([u_n]_{s,p}^p)
 -\frac{1}{p_s^{*}}M([u_n]_{s,p}^p)[u_n]_{s,p}^p
 +\Big(\frac{1}{ p} -\frac{1}{p^{*}_s}\Big)\|u_n\|_{p,V}^p\\
&\quad-  \beta \Big(\frac{1}{q}-\frac{1}{p^{*}_s} \Big)
 \int_{\mathbb{R}^N}k(x)|u_n|^q  \mathrm{d}x\\
&\ge\Big(\frac{1}{ p\theta} -\frac{1}{p^{*}_s}\Big)   M([u_n]_{s,p}^p)[u_n]_{s,p}^p
    +\Big(\frac{1}{ p} -\frac{1}{p^{*}_s}\Big)\|u_n\|_{p,V}^p\\
&\quad - \beta\Big(\frac{1}{q}-\frac{1}{p^{*}_s}\Big)\|k(x)\| _r
     \Big(\int_{\mathbb{R}^N} |u_n|^{p^*_s}  \mathrm{d}x \Big) ^{\frac{q}{p^*_s  }}  \\
&\ge\Big(\frac{1}{ p\theta} -\frac{1}{p^{*}_s}\Big)   m_0[u_n]_{s,p}^p
 +\Big(\frac{1}{ p} -\frac{1}{p^{*}_s}\Big)\|u_n\|_{p,V}^p \\
&\quad - \beta\Big(\frac{1}{q}-\frac{1}{p^{*}_s}\Big)\|k(x)\| _r
   S ^{-\frac{q}{p  }}  [u_n]_{s,p}^q\\
&\ge \min\Big\{\Big(\frac{1}{ p\theta} -\frac{1}{p^{*}_s}\Big)
 m_0,\Big(\frac{1}{ p} -\frac{1}{p^{*}_s}\Big)\Big\}\|u_n\|_{W}^p \\
&\quad - \beta\Big(\frac{1}{q}-\frac{1}{p^{*}_s}\Big)\|k(x)\| _r
   S ^{-\frac{q}{p  }} \|u_n\|_{W}^ {q}.
 \end{align*}
Since $\theta\in[1,\frac{N}{N-ps})$ and $q<p $,
 it follows that $\{u_n\}_n$ is bounded in $W $.
 Then, there exist $u_0$ and a subsequence, still denoted by
$\{u_n\}_n\subset W $, such that
\begin{gather*}
 u_n \rightharpoonup  u_0  \quad \text{weakly in }W , \\
  u_n \to  u_0 \quad \text{strongly in } L^t_{\rm loc}(\mathbb{R}^N)\quad
\text{for all } t\in [1,p^{*}_s),   \\
u_n \to u_0   \quad \text{a.e. in } \mathbb{R}^N\,.
\end{gather*}
From  Proposition \ref{p1.1}, we have
\begin{gather*}
u_n \rightharpoonup u_0 \quad \text{weakly in }   D^{s,p}(\mathbb{R}^N),\\
\int_{\mathbb{R}^{N}}\frac{|u_n(x)-u_n(y)|^{p}} {|x-y|^{N
 +ps}}\mathrm{d}y\rightharpoonup \mu \quad  \text{weakly $\ast$ in }
    \mathcal{\widetilde{M}}(\mathbb{R}^N),\\
|u_n(x)|^{p^*_s}\rightharpoonup \nu \quad  \text{weak $\ast$ in }
   \mathcal{\widetilde{M}} (\mathbb{R}^N).
\end{gather*}
Then
\begin{gather*}
\mu =\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}} {|x-y|^{N
 +ps}}\mathrm{d}y+\sum_{ j\in \mathcal{J}}\mu_j \delta_{x_j}+\tilde{\mu},\quad
 \mu(\mathbb{R}^N) \leq  C^p,\\
\nu = |u|^{p^*_s}+\sum_{ j\in \mathcal{J}}\nu_j \delta_{x_j},\quad
 \nu(\mathbb{R}^N) \leq S^{p^*_s} C^p,
\end{gather*}
where $J$ is at most countable, sequences
$\{\mu_j\}_j  , \{\nu_j\}_j  \subset \mathbb{R}^+_0$,
$\{x_j\}_j \subset \mathbb{R}^N$, $\delta_{x_j}$ is the Dirac mass centered
at $ x_j$,   $\tilde{\mu}$ is a non-atomic measure,
\begin{gather}\label{eqS2.9,}
\nu(\mathbb{R}^N) \leq S^{-\frac{p_{s}^*}{p}}\mu(\mathbb{R}^N)^{\frac{p_{s}^*}{p}}, \\
\label{eqS2.9,,}
\nu_j \leq S^{-\frac{p^*_s}{p}}\mu_j^{\frac{p_{s}^*}{p}},\quad \forall j \in J,
 \end{gather}
Concentration at infinity of the sequence $\{u_n\}_n$ is described by the
following quantities:
\begin{gather*}
\mu_{\infty}:
   =\lim_{R\to \infty} \limsup_{n\to \infty}
   \int_{\{x\in \mathbb{R}^N:|x|>R\}}
\int_{\mathbb{R}^N}\frac{ u_n(x)-u_n(y)| ^{p}}{|x-y|^{N+ps}}\mathrm{d}y\mathrm{d}x,
  \\
\nu_{\infty} :=\lim_{R\to \infty} \limsup_{n\to \infty}
\int_{\{x\in \mathbb{R}^N:|x|>R\}} |u_n| ^{p^*_s} \mathrm{d}x,
\end{gather*}

We claim that
$\mathcal{J}$ is finite and, for $j\in \mathcal{J }$,
either $\nu_j = 0$ or $\nu_j \ge (m_0 \alpha^{-1}S)^{N/ps}$.

     In fact, for $\varepsilon> 0$, let
 $\phi_j^{\varepsilon}(x)$ be a smooth cut-off function
 centered at $x_j$, such that
 $ 0\le \phi_j^{\varepsilon}(x)\le 1$,
  $\phi_j^{\varepsilon} (x)\equiv 0$ on
 $|x-x_j|\ge 2\varepsilon$, $\phi_j^{\varepsilon} (x)\equiv 1$ on
$|x-x_j|\le \varepsilon$, and $|\nabla \phi_j^{\varepsilon}(x)|\le  \frac{2}{\varepsilon}$
for all  $x\in \mathbb{R}^N$. Then, it is seen that $\{u_n\phi_j^{\varepsilon}\}$
is bounded   in $W $. Testing $I'(u_n)$ with
  $ u_n\phi_j^{\varepsilon} $, we obtain
  $\lim_{n\to \infty}\langle I'(u_n), \ u_n \phi_j^{\varepsilon} \rangle=0$;
   that is,
   \begin{equation}\label{eqS2.7'}
 \begin{aligned}
&M([u_n]_{s,p}^p) \iint_{\mathbb{R}^{2N}}
\Big(|u_n(x)-u_n(y)|^{p-2} (u_n(x)-u_n(y)) \\
&\times (u_n(x)\phi_j^{\varepsilon}(x)-u_n(y)\phi_j^{\varepsilon}(y))
/|x-y|^{N +ps}\Big)\mathrm{d}x\mathrm{d}y
+ \int_{\mathbb{R}^N}V(x)|u_n|^{p} \phi_j^{\varepsilon}\mathrm{d}x \\
&\quad -\alpha\int_{\mathbb{R}^N}|u_n|^{p_s^{*}}\phi_j^{\varepsilon}\mathrm{d}x 
 -\beta\int_{\mathbb{R}^N} k(x)|u_n|^q \phi_j^{\varepsilon}(x) \mathrm{d}x=0.
\end{aligned}
\end{equation}

Next we estimate each term in \eqref{eqS2.7'}.
\begin{equation}\label{eqS2.7',}
\begin{aligned}
&\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))
(u_n(x)\phi_j^{\varepsilon}(x)-u_n(y)\phi_j^{\varepsilon}(y))}{|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y\\
&=\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u_n(y)|^{p}
 \phi_j^{\varepsilon}(x) } {|x-y|^{N +ps}}\mathrm{d}x\mathrm{d}y\\
&\quad +\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))
 (\phi_j^{\varepsilon}(x)-\phi_j^{\varepsilon}(y)) u_n(y) } {|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y.
 \end{aligned}
\end{equation}
 In fact, in the first double
integral of the right-hand side of \eqref{eqS2.7'}, we can  use a compactness
result (see Proposition \ref{p1.1}),
\begin{equation}\label{eqS2.7'',}
\begin{aligned}
&\lim_{n\to \infty}\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)
 -u_n(y)|^{p} \phi_j^{\varepsilon}(x) } {|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y  \\
&=\int_{\mathbb{R}^N} \phi_j^{\varepsilon} \mathrm{d}\mu
=\iint_{\mathbb{R}^{2N}}\frac{|u_0(x)-u_0(y)|^{p}}
{|x-y|^{N +ps}}\phi_j^{\varepsilon}(x)\mathrm{d}x\mathrm{d}y +\mu_j.
\end{aligned}
\end{equation}
For the second double integral of the right-hand side of \eqref{eqS2.7',},
we obtain
\begin{equation}\label{eqS2.7'',b}
\begin{aligned}
&\Big|\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))
(\phi_j^{\varepsilon}(x)-\phi_j^{\varepsilon}(y)) u_n(y) } {|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y\Big|\\
&\leq \Big(\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u_n(y)|^{p}}  {|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y \Big)^{\frac{p-1}{p}}  \\
&\quad\times \Big(\iint_{\mathbb{R}^{2N}}\frac{|\phi_j^{\varepsilon}(x)
 -\phi_j^{\varepsilon}(y)|^p |u_n(y)|^p} {|x-y|^{N +ps}}\mathrm{d}x\mathrm{d}y
  \Big)^{1/p}\\
&\leq  C \Big(\iint_{\mathbb{R}^{2N}}\frac{|\phi_j^{\varepsilon}(x)
 -\phi_j^{\varepsilon}(y)|^p |u_n(y)|^p} {|x-y|^{N +ps}}\mathrm{d}x\mathrm{d}y
 \Big)^{\frac{p-1}{p}} .
 \end{aligned}
\end{equation}
By Lemma  \ref{lA.1}, we obtain
\begin{equation}\label{eqS2.7'7,,}
  \lim_{\varepsilon\to 0}\lim_{n\to \infty} \iint_{\mathbb{R}^{2N}}\frac{|\phi_j^{\varepsilon}(x)-\phi_j^{\varepsilon}(y)|^p |u_n(y)|^p} {|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y=0.
\end{equation}
So  from  \eqref{eqS2.7',}, \eqref{eqS2.7'',} and \eqref{eqS2.7'7,,}, we deduce
  \begin{equation}\label{eqS2.7'7,}
   \begin{aligned}
&\lim_{\varepsilon\to 0}\lim_{n\to \infty}  M[u_n]_{s,p}^p
 \iint_{\mathbb{R}^{2N}}\Big(|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y)) \\
&\times \big(u_n(x)\phi_j^{\varepsilon}(x)-u_n(y)\phi_j^{\varepsilon}(y)\big)/
|x-y|^{N +ps}\Big)\mathrm{d}x\mathrm{d}y\\
&\geq  m_0 \mu_j.
 \end{aligned}
\end{equation}
Also we have
 \begin{gather}\label{eqS2.10'',}
\lim_{\varepsilon\to 0}\lim_{n\to \infty} \int_{\mathbb{R}^N}V(x)  |u_n|^{p}
 \phi_j^{\varepsilon} \mathrm{d}x
=\lim_{\varepsilon\to 0}\lim_{n\to \infty}
 \int_{B_{2\varepsilon}(x_j)}V(x)|u_n|^{p}  \phi_j^{\varepsilon} \mathrm{d}x
=0, \\
\label{eqS2.10''}
 \lim_{n\to \infty} \int_{\mathbb{R}^N}  |u_n|^{p_{s}^*}   \phi_j^{\varepsilon}
\mathrm{d}x
 = \int_{\mathbb{R}^N}  \phi_j^{\varepsilon} \mathrm{d}\nu
= \int_{\mathbb{R}^N} |u_0|^{p_{s}^*}  \phi_j^{\varepsilon} \mathrm{d}x +  \nu_j .
   \end{gather}
 By  assumption (A4), we arrive at
\begin{equation}\label{eqS2.10}
\begin{aligned}
\lim_{\varepsilon\to 0}\lim_{n\to \infty}
\int_{\mathbb{R}^N}k(x)|u_n|^q\phi_j^{\varepsilon} \mathrm{d}x
&=  \lim_{\varepsilon\to 0}\lim_{n\to \infty}
\int_{B_{2\varepsilon}(x_j)}k(x)|u_n|^q\phi_j^{\varepsilon}  \mathrm{d}x
 \\
&\leq\lim_{\varepsilon\to 0}\lim_{n\to \infty}
\|k(x)\|_{L^r(B_{2\varepsilon}(x_j))}
\|u_n\|_{{L^{p_{s}^*}(B_{2\varepsilon}(x_j))}}^q \\
&=0.
\end{aligned}
 \end{equation}
Therefore, from \eqref{eqS2.7'} and the aforementioned  arguments we obtain
  \begin{equation}\label{eqS2.10b}
  0\geq m_0\mu_j-\alpha \nu_j.
\end{equation}
 Combining this with  \eqref{eqS2.9,,}, we
obtain either (i) $\nu_j = 0$ or
(ii) $\nu_j\ge (m_0 \alpha^{-1}S)^{\frac{N}{ps}}$,
which implies that $\mathcal{J}$ is finite. The claim is thereby
 proved.

To analyze the concentration at $\infty$,
 by choosing a suitable cut-off function
$\varphi\in C_{0}^{\infty}(\mathbb{R}^N,[0,1])$ such that
$\varphi(x)\equiv 0$ on $|x|\le 1$ and
$\varphi(x)\equiv 1$ on $|x|\ge 2$.
We set $\varphi_R(x)=\varphi(\frac{x}{R})$, then
$\{ u_n \varphi_R\}_n$ is bounded in
$W $, and $\lim_{n\to \infty}\langle I' (u_n), u_n \varphi_R\rangle=0 $; that is,
 \begin{equation}\label{eqS2.11}
   \begin{aligned}
&M([u_n]_{s,p}^p) \iint_{\mathbb{R}^{2N}}
\Big(|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y)) \\
&\times (u_n(x)\varphi_R(x)-u_n(y)\varphi_R (y))/
|x-y|^{N +ps}\Big)\mathrm{d}x\mathrm{d}y\\
&+ \int_{\mathbb{R}^N}V(x)|u_n|^{p} \varphi_R\mathrm{d}x  
   -\alpha\int_{\mathbb{R}^N}|u_n|^{p_s^{*}}\varphi_R\mathrm{d}x
-\beta\int_{\mathbb{R}^N} k(x)|u_n|^q \varphi_R \mathrm{d}x=0.
\end{aligned}
\end{equation}


Next we estimate each term in \eqref{eqS2.11}.
 \begin{equation}\label{eqS2.11,}
   \begin{aligned}
&\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))
(u_n(x)\varphi_R(x)-u_n(y)\varphi_R (y))}{|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y \\
&=\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u_n(y)|^{p} \varphi_R(x) } {|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y\\
&\quad +\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))
(\varphi_R(x)-\varphi_R(y)) u_n(y) } {|x-y|^{N  +ps}}\mathrm{d}x\mathrm{d}y.
 \end{aligned}
\end{equation}
 Indeed, in the first double
integral of the right-hand side of \eqref{eqS2.11,}, we can  use a
compactness result  (see Proposition \ref{p1.2}),
  \begin{equation}\label{eqS2.11,,}
\begin{aligned}
&\lim_{n\to \infty}\iint_{\mathbb{R}^{2N}}
\frac{|u_n(x)-u_n(y)|^{p}\varphi_R(x) } {|x-y|^{N +ps}}\mathrm{d}x\mathrm{d}y\\
&= \int_{\mathbb{R}^N}\varphi_R \mathrm{d}\mu\\
&= \iint_{\mathbb{R}^{2N}}\frac{|u_0(x)-u_0(y)|^{p}}  {|x-y|^{N
 +ps}}\varphi_R(x) \mathrm{d}x\mathrm{d}y   +  \mu_\infty.
 \end{aligned}
\end{equation}
For the second double integral of the right-hand side of \eqref{eqS2.11,},
it follows from the H\"older inequality that
\begin{equation}\label{eqS2.11,,,}
\begin{aligned}
&\Big|\iint_{\mathbb{R}^{2N}}\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))
 (\varphi_R(x)-\varphi_R(y)) u_n(y) } {|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y\Big| \\
&\leq  C \Big(\iint_{\mathbb{R}^{2N}}\frac{|\varphi_R(x)
 -\varphi_R(y)|^p |u_n(y)|^p} {|x-y|^{N +ps}}\mathrm{d}x\mathrm{d}y
 \Big)^{\frac{p-1}{p}} .
 \end{aligned}
\end{equation}
then, as in the proof of  \eqref{eqS2.7'7,,} we obtain
  \begin{equation*}
  \lim_{R\to\infty}   \limsup_{n\to\infty}
\iint_{\mathbb{R}^{2N}}\frac{|\varphi_R(x)-\varphi_R(y)|^p |u_n(y)|^p} {|x-y|^{N
 +ps}}\mathrm{d}x\mathrm{d}y=0.
 \end{equation*}
So  we have
  \begin{equation}\label{eqS2.7'7,b}
   \begin{aligned}
&\lim_{R\to\infty} \lim_{n\to \infty}M[u_n]_{s,p}^{p}\iint_{\mathbb{R}^{2N}}
\Big(|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))\\
&\times (u_n(x)\varphi_R(x)-u_n(y)\varphi_R(y))/|x-y|^{N +ps}\Big)
\mathrm{d}x\mathrm{d}y\\
&\geq m_0  \mu_{\infty}.
 \end{aligned}
\end{equation}
We also obtain
 \begin{gather}\label{eqS2.10'',b}
\lim_{R\to\infty} \lim_{n\to \infty} \int_{\mathbb{R}^N} V(x) |u_n|^{p}
 \varphi_R \mathrm{d}x
= \lim_{R\to\infty} \lim_{n\to \infty} \int_{\{|x|>2R\}}V(x)  |u_n|^{p}
\varphi_R \mathrm{d}x =0, \\
\label{eqS2.10''b}
\lim_{n\to \infty} \int_{\mathbb{R}^N}  |u_n|^{p_{s}^*}   \varphi_R \mathrm{d}x
 = \int_{\mathbb{R}^N}  \varphi_R  \mathrm{d}\nu
= \int_{\mathbb{R}^N} |u_0|^{p_{s}^*}   \varphi_R \mathrm{d}x +  \nu_{\infty} .
 \end{gather}
By the weak continuity of $k(x)$, H\"older inequality and the definition of $S$,
  \begin{align*}
\int_{\mathbb{R}^N}k(x)|u_n|^q\varphi_R \mathrm{d}x
 &\le \Big(\int_{\{|x|>2R\}} |u_n|^{p_s^{*}}  \mathrm{d}x\Big)^\frac{q}{p_s^{*}}
\Big(\int_{\{|x|>2R\}}|k(x)| ^{\frac{p_s^{*}}{p_s^{*}-q} }
\mathrm{d}x\Big)^\frac{p_s^{*}-q}{p_s^{*}}
    \\
&\le  S^{-\frac{q}{p_s^{*}}}[u_n]_{s,p}^q
 \Big(\int_{\{|x|>2R\}}|k(x)| ^\frac{p_s^{*}}{p_s^{*}-q}
 \mathrm{d}x\Big)^\frac{p_s^{*}-q}{p_s^{*}}\\
&\le  S^{-\frac{q}{p_s^{*}}}\|u_n\|_{W}^q
 \Big(\int_{\{|x|>2R\}}|k(x)| ^\frac{p_s^{*}}{p_s^{*}-q}
\mathrm{d}x\Big)^\frac{p_s^{*}-q}{p_s^{*}},
\end{align*}
which implies
$$
  \lim_{R\to\infty} \limsup_{n\to\infty} \int_{\mathbb{R}^N}k(x)|u_n|^q\varphi_R
\mathrm{d}x
\le  C  \lim_{R\to\infty}   \Big(\int_{\{|x|>2R\}}|k(x)| ^\frac{p_s^{*}}{p_s^{*}-q}
 \mathrm{d}x\Big)^\frac{p_s^{*}-q}{p_s^{*}}=0.
$$
Therefore, by letting $R\to\infty$ and $n\to\infty$ in \eqref{eqS2.11},
we have
  \begin{equation} \label{eqS2.11.}
m_0\mu_{\infty } \le  \alpha \nu_{\infty }.
\end{equation}
By Proposition \ref{p1.2} and \eqref{eqS2.11.}, we conclude that
either 
\begin{itemize} 
\item[(iii)] $\nu_{\infty} = 0$, or
\item[(iv)] $\nu_{\infty}\ge (m_0\alpha^{-1}S )^{\frac{N}{ps}}$.
\end{itemize}

Next, we claim that  (ii)  and  (iv)  cannot occur if $\alpha$ and  $\beta$
are chosen properly.  To this, from the H\"older inequality and the weak
continuity of $\mathcal{F}$, we have
\begin{equation}\label{eqS2.18,}
\begin{aligned}
0>c
&= \lim_{n\to \infty} \Big[ I(u_n)-\frac{1}{p_s^{*}} \langle I'(u_n),  u_n  \rangle \Big]
 - \beta \Big(\frac{1}{q}-\frac{1}{p^{*}_s} \Big)\int_{\mathbb{R}^N}k(x)|u_n|^q  \mathrm{d}x\Big]\\
 &\ge \Big(\frac{1}{ p\theta} -\frac{1}{p^{*}_s}\Big)   M([u_0]_{s,p}^p)[u_0]_{s,p}^p+\Big(\frac{1}{ p} -\frac{1}{p^{*}_s}\Big)   \|u_0\|_{p,V}^p\\
 &\quad -  \beta\Big(\frac{1}{q}-\frac{1}{p^{*}_s}\Big)\|k(x)\| _r
    \Big (\int_{\mathbb{R}^N} |u_0|^{p^*_s}  \mathrm{d}x \Big) ^{\frac{q}{p^*_s  }}  \\
&\ge\Big(\frac{1}{ p\theta} -\frac{1}{p^{*}_s}\Big)    m_0[u_0]_{s,p}^p
     +\Big(\frac{1}{ p} -\frac{1}{p^{*}_s}\Big) \|u_0\|_{p,V}^p \\
&\quad - \beta\Big(\frac{1}{q}-\frac{1}{p^{*}_s}\Big)\|k(x)\| _r
   S ^{-\frac{q}{p  }} [u_0]_{s,p}^{\frac{q}{p}} \\
&\ge \Big(\frac{1}{p\theta}-\frac{1}{p^{*}_s}\Big)m_0 S\|u_0\|_{p^*_s }^p
 - \beta\Big(\frac{1}{q}-\frac{1}{p^{*}_s}\Big)\|k(x)\| _r
   \|u_0\|_{p^*_s }^q.
   \end{aligned}
 \end{equation}
Thus, it follows that
    \begin{equation}\label{eqS2.19,}
\|u_0\|_{p^*_s }\leq C \beta^\frac{1}{p-q}.
   \end{equation}
If (iv)  occurs, we obtain by  \eqref{eqS2.19,}  that
\begin{align*}
0>c
&= \lim_{R\to \infty}\lim_{n\to \infty}
 \Big[ I(u_n)-\frac{1}{p_s^{*}} \langle I'(u_n),  \varphi_R  \rangle \Big]\\
&\ge\Big(\frac{1}{ p\theta} -\frac{1}{p^{*}_s}\Big)   m_0\mu_{\infty}
 +\Big(\frac{1}{ p} -\frac{1}{p^{*}_s}\Big)   \|u_0\|_{p,V}^p
 - \beta\Big(\frac{1}{q}-\frac{1}{p^{*}_s}\Big)\|k(x)\| _r   \|u_0\|_{p^*_s }^q  \\
&\ge\Big(\frac{1}{ p\theta} -\frac{1}{p^{*}_s}\Big)    m_0\mu_{\infty}
   - \beta\Big(\frac{1}{q}-\frac{1}{p^{*}_s}\Big)\|k(x)\| _r  C \beta^\frac{q}{p-q}\\
&\ge  \Big(\frac{1}{p\theta}-\frac{1}{p^{*}_s}\Big)m_0
 \alpha^{-\frac{N}{ps}}S^{\frac{N}{ps}}-C \beta^{\frac{p }{p-q}}\,.
\end{align*}
However, since $\theta \in [1, \frac{N}{N-ps}), q<p$, if  $\alpha> 0$ is given,
we can take small  $\beta_*$   such that for every
$0 <\beta< \beta_*$, the  term on the right-hand side above is greater than zero,
 which is a contradiction. Similarly, if $\beta> 0$ is given, we can choose
small $\alpha_*$ such that for every $0 <\alpha <\alpha_*$, the   term on the
right-hand side above is greater than zero. Similarly, we can prove that (ii)
 cannot occur.  Hence
$$
 \int_{\mathbb{R}^N} |u_n|^{p^{*}_s}\mathrm{d}x  \to
 \int_{\mathbb{R}^N}|u_0|^{p^{*}_s}\mathrm{d}x   \quad \text{as } n\to \infty.
$$
 In view of $u_n \rightharpoonup  u_0$ in $W^{s,p}(\mathbb{R}^N)$ and
the Br\'{e}zis-Lieb lemma, we have
  \begin{equation} \label{eqS2.,,11.}
 \int_{\mathbb{R}^N} |u_n-u_0|^{p^{*}_s}\mathrm{d}x  \to
0  \quad \text{as } n\to \infty.
\end{equation}
We are now in a position to show that $\{u_n\}_n$   converges strongly
to $u_0$ in $W $. Firstly, we have
$$
\langle I'(u_n)-I'(u_0),
u_n-u_0\rangle \to 0 \quad  \text{as } n\to  \infty.
$$
By the boundedness of $\{u_n\}_n$ in $W $ and \eqref{eqS2.,,11.}, it follows that
   \begin{equation} \label{eqS2.,11.}
\begin{aligned}
&\int_{\mathbb{R}^N} (|u_n|^{p^{*}_s-2}|u_n|-|u_0|^{p^{*}_s-2}u_0)(u_n-u_0)
 \mathrm{d}x  \\
& \leq \int_{\mathbb{R}^N} |u_n|^{p^{*}_s-1}( u_n - u_0) \mathrm{d}x
 + \int_{\mathbb{R}^N} |u_0|^{p^{*}_s-1}( u_n - u_0) \mathrm{d}x\\
& \leq \Big(\int_{\mathbb{R}^N}  |u_n|^{p^{*}_s}
 \mathrm{d}x\Big)^{\frac{p^{*}_s-1}{p^{*}_s}}
 \Big(\int_{\mathbb{R}^N}  |u_n - u_0|^{p^{*}_s } \mathrm{d}x\Big)^{1/p^*_s}\\
&\quad +\Big(\int_{\mathbb{R}^N}  |u_0|^{p^{*}_s}\mathrm{d}x
 \Big)^{\frac{p^{*}_s-1}{p^{*}_s}}
 \Big(\int_{\mathbb{R}^N}  |u_n - u_0|^{p^{*}_s } \mathrm{d}x\Big)^{1/p^*_s}
\to 0 ,
\end{aligned}\end{equation}
as $n\to  \infty$. Since $k\in L^r(\mathbb{R}^N)$, by the weak lower
continuity of   $\mathcal{F}$, we have
$$
 \int_{\mathbb{R}^N}k(x)|u_n|^{q-2}u_n(u_n-u_0)  \mathrm{d}x
\leq \|k(x)\|_{L^r(\mathbb{R}^N)} \|u_n\|_{p^{*}_s}^{q-1} \|u_n - u_0\|_{p^{*}_s }
\to 0,
$$
as $n \to  \infty$.
Therefore, as $n \to  \infty$, we have
\begin{equation*}
   [u_n-u_0]_{s,p}\to 0,
\end{equation*}
thanks to $I'(u_0) = 0$.
 By Lemma \ref{lemma4.6},  as $n\to  \infty$, we have
$$
\int_{\mathbb{R}^N} V(x)  |u_n|^{p-2}u_n(u_n-u_0)   \mathrm{d}x\to   0.
$$
Thus we prove that $\{u_n\}_n$ strongly converges to $u_0$ in $W$.
\end{proof}

\section{Proof of Theorem \ref{T1.1}}

In this section, we use minimax procedure (see \cite{PHR}) to prove
the existence of infinitely many solutions.  Let   $X$ be a Banach space
and $\Sigma$ be the class of subsets of
$X\setminus \{0\}$ which are closed and symmetric with
respect to the origin. For $A \in \Sigma$, we define the genus
$ \gamma(A)$ by
\begin{gather*}
\gamma(A) = \inf\{n \in  \mathbb{N}: \exists\phi
\in C(A, \mathbb{R}^n \setminus \{0\}), \phi(z)=-\phi(-z)\}, \\
N_{\delta} (A) =
\{x\in X :  \operatorname{dist}(x- A)\le \delta\},\quad
\operatorname{dist}(x- A)=\inf\{\|x- A\|: y\in A\}.
\end{gather*}
If there is no mapping as above for any $n \in \mathbb{N}$, then $\gamma (A) = +\infty$.
 Let $\Sigma_n$ denote the family of closed symmetric subsets $A$
of $X$ such that $0\not\in A$ and $ \gamma(A)\ge n$. We summarize the property
of genus, which will be used in the proof of Theorem \ref{T1.1}.
 We refer the readers to \cite{PHR} for the proof of the next result.

\begin{proposition}\label{p3.3}
 Let $A$ and $B$ be closed symmetric subsets of $X$ which do not contain
the origin. Then the following hold.
\begin{itemize}
\item[(1)] If there exists an odd continuous mapping from $A$ to $B$,
then $ \gamma (A)\le \gamma  (B)$;

\item[(2)] If there is an odd homeomorphism from $A $ to $B$, then
$\gamma (A) =\gamma (B)$;

\item[(3)] If $\gamma  (B) <\infty$, then
 $\gamma (\overline{A\backslash B}) \ge \gamma  (A) -\gamma (B)$;

\item[(4)]  $n$-dimensional sphere $S_n$ has a genus of $n + 1$
by the Borsuk-Ulam Theorem;

\item[(5)] If $A$ is compact, then $\gamma (A) < +\infty$ and there exists
$ \delta> 0$ such that  $N_{\delta} (A)\subset\Sigma $ and
$\gamma (N_{\delta} (A)) =\gamma (A)$.
\end{itemize}
\end{proposition}

 The following version of the symmetric mountain-pass lemma is due to Kajikiya
 \cite{KR}.

\begin{proposition}\label{p3.1}
Let $E$ be an infinite-dimensional space and
$I\in C^1(E, \mathbb{R})$ and suppose the following conditions hold.
\begin{itemize}
\item[(A5)] $I(u)$ is even, bounded from below,  $I(0) = 0$  and $I(u)$
   satisfies the local Palais-Smale condition (PS for short).

\item[(A6)]  For each  $n \in \mathbb{N}$,  there exists an
   $A_n\in \Sigma_n $  such that   $\sup_{u\in A_n} I(u) < 0$.
\end{itemize}
Then either 
\begin{itemize}
\item[(i)]  There exists a sequence  $\{u_n\}$  such that
     $I'(u_n) = 0, I(u_n) < 0 $  and $\{u_n\}$  converges to zero, or

\item[(ii)]  There exist two sequences $\{u_n\}$  and $\{v_n\}$   such that
 $ I'(u_n) = 0, I(u_n) = 0$, $u_n  \neq 0$, $\lim_{n\to \infty} u_n = 0$;
$I'(v_n) = 0$, $I(v_n) <0$, $\lim_{n\to \infty}I(v_n) = 0$,  and
$\{v_n\}$  converges to a non-zero limit.
\end{itemize}
 \end{proposition}

 \begin{remark}\label{r3.1} \rm
From Proposition \ref{p3.1} we have a sequence
   $\{u_n\}_n$ of critical points such that  $I(u_n)\le 0, u_n\neq 0 $ and
$\lim_{n\to \infty}u_n = 0$.
 \end{remark}

To obtain infinitely many solutions, we need some technical lemmas.
Let  $I(u) $ be the functional defined as above,  $1 < q <  p$,
$\alpha > 0$ and $\beta > 0$. Then
\begin{align*}
 I(u)  &= \frac{1}{p}\big[
\mathcal{M}( [u]_{s,p}^p)+\|u\|_{p,V}^p\big]
    -\frac{\alpha}{p^{*}_s}\int_{\mathbb{R}^N}
|u|^{p^{*}_s}\mathrm{d}x-\frac{\beta}{q}\int_{\mathbb{R}^N}
k(x)|u|^q\mathrm{d}x
 \\
&\ge \frac{1}{p\theta}{M}( [u]_{s,p}^p) [u]_{s,p}^p+\frac{1}{p}\|u\|_{p,V}^p
-\frac{\alpha}{p^{*}_s}\int_{\mathbb{R}^N}
|u|^{p^{*}_s}\mathrm{d}x-\frac{\beta}{q}\int_{\mathbb{R}^N}
k(x)|u|^q\mathrm{d}x
 \\
&\ge \frac{1}{p\theta}m_0 [u]_{s,p}^p
 - \frac{\alpha}{p^{*}_s}\int_{\mathbb{R}^N} |u|^{p^{*}_s}  \mathrm{d}x
- \frac{\beta}{q} \|k(x)\|_r \|u\|_{p^{*}_s}^q
 \\
 &\ge  \frac{1}{p\theta}m_0 [u]_{s,p}^p
 -\frac{\alpha}{p^{*}_s} \Big(S^{-1} [u]_{s,p}^p\Big)^{\frac{p^{*}_s}{p}}
- \frac{\beta}{q}\|k(x)\|_r \Big(S^{-1} [u]_{s,p}^p\Big)^{\frac{q}{p}}
  \\
 &\ge C_1 [u]_{s,p}^p-\alpha C_2 [u]_{s,p}^{p^{*}_s} -\beta C_3 [u]_{s,p}^q .
 \end{align*}
Define
$$
g(t)= C_1t^p -\alpha C_2t^{p^{*}_s}-\beta C_ 3t^q .
$$
Then, it is easy to see that,
for the given $\alpha > 0$, we can choose $\beta^*> 0$ so small that
if $0 < \beta < \beta^*$, there exists
$0 < t_0 < t_1 $   such that $g(t) < 0$ for  $0 < t < t_0$;
 $g(t) > 0$ for $t_0 < t < t_1$; $g(t) < 0$ for $t> t_1$.

Similarly, for the given $\beta> 0$, we can choose $\alpha^*> 0$ so small that
 if $0 < \alpha < \alpha^*$, there exists
$0 < t_0^* < t_1^* $   such that $g(t) < 0$ for
$0 < t < t_0^*;$ $g(t) >0$ for $t_0^* < t < t_1^*$;
 $g(t) < 0$ for $t > t_1^*$.

Clearly, $g(t_0) = 0 = g(t_1)$. Following the same
idea as in \cite{FJI}, we consider the truncated functional
\begin{equation*}
 \tilde{I}(u)  = \frac{1}{p}\Big[
\mathcal{M}( [u]_{s,p}^p)+\|u\|_{p,V}^p\Big]
- \frac{\alpha}{p^{*}_s}\psi(u)\int_{\mathbb{R}^N}   |u|^{p^{*}_s}  \mathrm{d}x
- \frac{\beta}{q}\int_{\mathbb{R}^N}  k(x)|u|^q\mathrm{d}x.
\end{equation*}
where $\psi(u) =\tau(\|u\|_W)$ and
$\tau: \mathbb{R}^+ \to [0,1]$ is a non-increasing $C^\infty$
function such that $\tau(t) = 1$
if $t\le t_0$ and $\tau(t) = 0$ if $t\ge t_1$. Obviously,
$\tilde{I}(u)$ is even. Thus, from Lemma \ref{l2.5}, we obtain the
following lemma.


 \begin{lemma}\label{l2.5'}
 Let $ c < 0$ and $1 <q< p$. Then
\begin{itemize}
\item[(1)] $\tilde{I}\in  C^1$ and $\tilde{I}$ is bounded from below.

\item[(2)] If $\tilde{ I}(u) < 0$, then  $\|u\|_W< t_0$ and $\tilde{I}(u) = I(u)$.


\item[(3)]   for each $\alpha > 0$ there exists
$\widetilde{\beta^*}=\min\{\beta_*,\beta^*\}> 0$ such that if $0 <\beta< \widetilde{\beta^*}$,
 then $\tilde{I}$ satisfies $(PS)_c$;

\item[(4)] for each $\beta> 0$ there exists
$\widetilde{\alpha^*}=\min\{\alpha_*, \alpha^*\} > 0$ such that if $0 < \alpha
< \widetilde{\alpha^*}$, then $ \widetilde{I}$ satisfies $(PS)_c$.
\end{itemize}
\end{lemma}

\begin{proof} Obviously, (1) and (2) are immediate. To prove (3) and (4),
 observe that all $(PS)_c$ sequences for $\tilde{I}$ with $c < 0$
must be bounded, similar to the proof of Lemma \ref{l2.5}, there
exists a strong convergent subsequence in $W^{s,p}(\mathbb{R}^N )$.
\end{proof}

\begin{remark}\label{r3.2}\rm
 Denote $K_c =\{u\in W:  \tilde{I}'(u) = 0,\tilde{I}(u) = c\}$
  If $\alpha, \beta$ are as in $(3)$ or $(4)$ above, then, it follows from $(PS)_c$ that
$K_c (c < 0)$ is compact.
\end{remark}

\begin{lemma}\label{l3.3}
 Denote $\tilde{I}_c: = \{u\in W :  \tilde{I}'(u) = 0,\tilde{I}(u) \leq c\}$.
Given $n \in \mathbb{N}$, there exists $\epsilon_n < 0$, such that
$$
\gamma(\tilde{I}^{\epsilon_n}):=\gamma(  \{u\in W :
\tilde{I}(u) \le \epsilon_n\})\ge n.
$$
\end{lemma}

\begin{proof}
 Let $X_n$ be a $n$-dimensional subspace of $W $. For any
 $ u\in X_n,   u \not= 0$, write $ u  =r_{n}w $
with $w \in   X_n, \|w \|_W= 1$ and then $r_{n}=\|u\|_W  $.
 From the assumptions  $k(x)$, it is easy to see that, for
every $w\in X_n$  with  $\|w\|_W = 1$, there
exists $d_n > 0$ such that $ \int_{\mathbb{R}^N } k(x)
|w|^q  \mathrm{d}x\ge d_n$. Thus for $0 <r_n < t_0$, by the continuity of $M$, we
have
\begin{align*}
\tilde{I}(u)&=  \frac{1}{p}\Big[
\mathcal{M}([u]_{s,p}^p)+\|u\|_{p,V}^p\Big]
- \frac{\alpha}{p^{*}_s}\psi(u)\int_{\mathbb{R}^N}   |u|^{p^{*}_s}  \mathrm{d}x
- \frac{\beta}{q}\int_{\mathbb{R}^N}  k(x)|u|^q\mathrm{d}x
 \\
&\le  \frac{1}{p}r_n^p\Big[
\mathcal{M}([w]_{s,p}^p)+\|w\|_{p,V}^p\Big]
- \frac{\alpha}{p^{*}_s} r_n^{p^{*}_s}\int_{\mathbb{R}^N}   |w|^{p^{*}_s}
  \mathrm{d}x
  - \frac{\beta}{p}r_n^q \int_{\mathbb{R}^N}  k(x)|w|^q\mathrm{d}x   \\
 &\le   \frac{C_1}{p}r_n^p%\Big[\mathcal{M}([w]_{s,p}^p)+\|w\|_p^p\Big]
   - \frac{\alpha}{p^{*}_s} r_n^{p^{*}_s}\int_{\mathbb{R}^N}   |w|^{p^{*}_s}
  \mathrm{d}x
- \frac{\beta}{p}  d_nr_n^q \\
& =\epsilon_n.
\end{align*}
Therefore, we can choose $r_n\in (0, t_0)$ so
small that $\tilde{I}(u)\le \epsilon_n < 0$. Let
\begin{equation}\label{eqS3.1}
S_{r_n} = \{u\in X_n: \|u\|_W =r_n\}.
\end{equation}
Then $S_{r_n}  \cap X_n\subset \tilde{I}^{\epsilon_n}$.
 Hence by Proposition \ref{p3.3},
$$
\gamma(\tilde{I}^{\epsilon_n} )\ge \gamma(S_{r_n} \cap X_n) =n.
$$
As desired.
\end{proof}

According to Lemma \ref{l2.5'}
we  denote $\Sigma_n = \{A\in \Sigma:\gamma (A)\ge n\}$ and let
\begin{equation}\label{eqS3.1'}
c_n = \inf_{A\in \Sigma_n}\sup_{u\in A}\tilde{I}(u).
\end{equation}
Then
\begin{equation}\label{eqS3.1,'}
-\infty< c_n\le \epsilon_n <0
\end{equation}
 because $\tilde{I}^{\epsilon_n}\in \Sigma_n $ and $\tilde{I}$ is bounded
from below.

\begin{lemma}\label{l3.4}
 Let $\alpha,  \beta$ be as in $(3)$ or $(4)$ of Lemma \ref{l2.5'}.
Then all $c_n$ (given by  \eqref{eqS3.1'}) are critical values of
$\tilde{I}$ and $c_n \to 0$.
\end{lemma}

\begin{proof}
Since $\Sigma_{n+1}\subset \Sigma_n $, it is clear that $c_n\le c_{n+1}$.
By  \eqref{eqS3.1,'},  we have $c_n < 0$. Hence there is a
$  \bar{c}\le 0$ such that $c_n\to \bar{c}\le 0$. Moreover, since that all
$c_n$ are critical values of $\tilde{I}$ (see \cite{PHR}), we claim that
$\bar{c} = 0$. If $ \bar{c} < 0$,
then by Remark \ref{r3.2},
$K_{\bar{c}}=\{u\in W: \tilde{I}'(u)=0,\tilde{I}(u)=\bar{c}\}$ is compact and
$K_{\bar{c}}\in \Sigma$, then
$\gamma (K_{\bar{c}}) = n_0 <+\infty$ and there
exists $\delta>0$ such
 that $\gamma (K_{\bar{c}}) = \gamma (N_\delta(K_{\bar{c}}) ) =n_0$,
 here $N_{\delta}(K_{\bar{c}}) = \{x \in X:  \|x- K_{\bar{c}}\|\le \delta\}$.
By the deformation lemma (see \cite{M1}), there exist $\epsilon > 0$
$(\bar{c} +\epsilon < 0)$ and an odd homeomorphism $\eta:W\to W$ such that
$$
\eta(\tilde{I}^{\bar{c} +\epsilon }\setminus N_\delta(K_{\bar{c}}))
\subset \tilde{I}^{\bar{c}-\epsilon }.
$$
Since $c_n$ is increasing and converges to $\bar{c}$, there exists
$n\in \mathbb{N}$ such that $c_n > \bar{c}-\epsilon$  and
$c_{n +n_0}\le \bar{c}$. Choose $A\in  \Sigma_{n+n_0}$ such that
$\sup_{u\in A} \tilde{I}(u)< \bar{c} +\epsilon$, that is
$A\subset \tilde{I}^{\bar{c} +\epsilon}$.
 By the properties of $\gamma$, we have
$$
\gamma (\overline{A \setminus N_\delta(K_{\bar{c}})}) \ge
\gamma(A)-\gamma(N_\delta(K_{\bar{c}}))) \ge n, \ \  \gamma
(\overline{\eta(A \setminus N_\delta(K_{\bar{c}}))})\ge n.
$$
Hence, we have $\overline{\eta(A \setminus N_\delta(K_{\bar{c}}))}\in
\Sigma_n$. Consequently,
\[
 \sup_{u\in\overline{\eta(A \setminus
N_\delta(K_{\bar{c}}))}} \tilde{I}(u) \ge c_n > \bar{c}-\epsilon,
\]
a contradiction, hence $c_n\to 0$.
\end{proof}

\begin{proof}[Proof of Theorem  \ref{T1.1}]
By Lemma \ref{l2.5'}(2),  $\tilde{I} (u) = I(u)$ if  $\tilde{I} (u)< 0$.
Then, by Lemmas \ref{l2.5'}--\ref{l3.4}, one can see that all the assumptions
of Proposition \ref{p3.1} are satisfied. This completes the proof.
\end{proof}


\subsection*{Acknowledgments}
 L. Wang was supported by the  NSFC (11561024,11326139),
Youth Science Foundation program of Jiangxi Provincial (20151BAB201017)
and  the science and technology project of Jiangxi provincial education
 department (GJJ150537). B. Zhang was supported by the Natural Science
 Foundation of Heilongjiang Province of China (No. A201306)
and the Research Foundation of Heilongjiang Educational Committee (No. 12541667).


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