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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 208, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/208\hfil Existence of infinitely many solutions]
{Existence of infinitely many solutions for fractional $p$-Laplacian equations
with sign-changing potential}

\author[Y. Zhang, X. Tang, J. Zhang \hfil EJDE-2017/208\hfilneg]
{Youpei Zhang, Xianhua Tang, Jian Zhang }

\address{Youpei Zhang \newline
School of Mathematics and Statistics,
Central South University,
Changsha, Hunan 410083, China}
\email{zhangypzn@163.com}

\address{Xianhua Tang \newline
School of Mathematics and Statistics,
Central South University,
Changsha, Hunan 410083, China}
\email{tangxh@mail.csu.edu.cn}

\address{Jian Zhang (corresponding author) \newline
School of Mathematics and Statistics,
Hunan University of Commerce,
Changsha, 410205 Hunan, China.\\
Key Laboratory of Hunan Province for Mobile Business Intelligence,
Hunan University of Commerce,
Changsha, 410205 Hunan, China}
\email{zhangjian433130@163.com}

\dedicatory{Communicated by Giovanni Molica Bisci}

\thanks{Submitted January 25, 2017. Published September 8, 2017.}
\subjclass[2010]{35J60, 35J20}
\keywords{Fractional $p$-Laplacian; multiple solutions; variational methods;
\hfill\break\indent  sign-changing potential}

\begin{abstract}
 In this article, we prove the existence of infinitely many solutions
 for the  fractional $p$-Laplacian equation
 $$
  (-\Delta)^s_p u+V(x)|u|^{p-2}u=f(x,u),\quad x\in \mathbb{R}^N
 $$
 where $s\in(0,1)$, $2\leq p<\infty$. Based on a direct sum decomposition
 of a space $E^s$, we investigate the multiplicity of solutions
 for the fractional $p$-Laplacian equation in $\mathbb{R}^N$. The potential
 $V$ is allowed to be sign-changing, and the primitive of the nonlinearity
 $f$ is of super-$p$ growth near infinity in $u$ and allowed to be sign-changing.
 Our assumptions are suitable and different from those studied previously.
\end{abstract}

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

\section{Introduction and statement of main results}

We consider the  fractional $p$-Laplacian equation
\begin{equation}\label{q}
 (-\Delta)^s_p u+V(x)|u|^{p-2}u=f(x,u),\quad x\in {\mathbb{R}^N},
\end{equation}
where $(-\Delta)^s_p$ denotes the fractional $p$-Laplacian operator,
$0<s<1$, $2\leq p<\infty$, $f: \mathbb{R}^N\times\mathbb{R}\mapsto\mathbb{R}$,
$V: \mathbb{R}^N\mapsto\mathbb{R}$, $f$ and $V$ are allowed to be sign-changing.
Equation \eqref{q} driven by the fractional Laplacian arises in various
areas and different applications, such as phase transitions, finance,
stratified materials, flame propagation, ultra-relativistic limits of
quantum mechanics, and water waves. For more detailed introductions
and applications, we refer the reader to \cite{AS, NL}.
With the aid of variational methods, the existence and
multiplicity of nontrivial solutions for  \eqref{q} have been extensively
investigated in the literature over the past several
decades. See e.g., \cite{CM, DI,11DONG,FP,X7,SE,SEO,TE,10XU} and the references
quoted in them.

For $p=2$,   \eqref{q} reduces to the so-called fractional Schr\"{o}dinger equation
\begin{equation}\label{1}
(-\Delta)^s u+V(x)u=f(x,u), \quad x\in\mathbb{R}^N.
\end{equation}
Equation  \eqref{1} arises in the study of the nonlinear fractional
 Schr\"{o}dinger equation
\begin{equation*}
  i\frac{\partial\varphi }{\partial t}
=(-\Delta)^s\varphi+W(x)\varphi-f(x,|\varphi|)\varphi,\quad
(t,x)\in {\mathbb{R}\times\mathbb{R}^N}.
\end{equation*}
 Most of the references deal with the
  \eqref{1} with the potential $\inf_{x\in {\mathbb{R}^N}}V(x)>0$,
and the others handled the case where $V(x)$ is sign-changing. For the second
case, the classical proof is based on the following classical condition
which was introduced by Ambrosetti and Rabinowitz in \cite{T26}:
(AR) there exists $\mu>2$ such that
\begin{equation*}
  0<\mu F(x,t)\leq t f(x,t),\quad t\neq0,
\end{equation*}
where $F(x,t)=\int^t_0f(x,\tau) d\tau$. In \cite{SE}, the authors used
the concentration compactness principle to show that   \eqref{1}
$(V(x)\equiv1)$ has at least two nontrivial radial solutions without the (AR)
 condition. In \cite{11DONG,TE,10XU}, the authors used variant Fountain Theorems
and the $\mathbb{Z}_2$ version of Mountain Pass Theorem to establish some
new existence theorems on infinitely many nontrivial high or small energy
solutions for \eqref{1}.

 Furthermore, there are also many classical and fantastic studies on nonlocal
fractional problems, see for example,
 \cite{18CH,17CH,15HU,14TA,GB3,GM2,GM1,GB4,12SH,13Sh,ZB,ZTZ,ZZM}. In \cite{GM2}, the authors study the existence of multiple ground state solutions for a class of parametric fractional Schr\"{o}dinger equations. In \cite{ZB}, the authors obtain the existence of infinitely many weak solutions for
equations driven by nonlocal integrodifferential operators with homogeneous
Dirichlet boundary conditions. Recently, some authors have been concerned about
the general case (i.e., $p\neq2$)  of   \eqref{q}, see for example
\cite{CHEN,21X8,TO,XJ}. In \cite{TO}, by using Mountain Pass Theorem
with Cerami condition, Torres established the existence of weak solutions for
 \eqref{q}, in which the nonlinearity $f(x,u)$ is subcritical and $p$-superlinear,
and the potential $V(x)$ satisfies coercive condition at infinity.
For general case $p>2$, if $V(x)$ is a sign-changing potential,   \eqref{q}
is far more difficult as $((-\Delta)^s_p+V)$ is no longer a self-adjoint and
so a complete description of its spectrum is not available.
For these reasons, only a few papers have treated this case so far.
 In \cite{CHEN}, by applying Mountain Pass Theorem, Cheng considered this
case and established the existence of one nontrivial solutions for   \eqref{q}.

In a recent paper \cite{VA}, Vincenzo ambrosio used a variant of the Fountain
Theorem to prove the existence of infinitely many nontrivial weak solutions for
  \eqref{q}, where the following the assumptions on $V$ and $f$ are introduced:
\begin{itemize}
\item[(A1)] $V\in{C(\mathbb{R}^N,{\mathbb{R}})}$ and 
$\inf_{x\in\mathbb{R}^N}V(x)>-\infty$;

\item[(A2)] there exists a constant $d_0>0$ such that
\begin{equation*}
  \lim_{|y|\to +\infty}\operatorname{meas}\{x\in{\mathbb{R}^N}:
 |x-y|\leq d_0,V(x)\leq M\}=0,\quad \forall  M>0,
\end{equation*}
where $\operatorname{meas}(\cdot)$ denotes the Lebesgue measure in $\mathbb{R}^N$;

\item[(A3)] $f\in C({\mathbb{R}^N},{\mathbb{R}})$, and there exist constant $c_1,c_2>0$
and $q\in {(p,p^*_s)}$ such that
\begin{equation*}
  |f(x,t)|\leq c_1|t|^{p-1}+c_2|t|^{q-1},\quad \forall (x,t)\in 
\mathbb{R}^N\times\mathbb{R},
\end{equation*}
where $p^*_s=\infty$ if $N\leq  sp$ and $p^*_s=\frac{Np}{N-sp}$ 
if $N> sp$, $p^*_s$ is the fractional critical exponent;

\item[(A4)] there exists $\sigma\geq1$ such that
\begin{equation*}
  \sigma \mathcal{F}(x,t)\geq \mathcal{F}(x,\tau t),\quad
\forall (x,t)\in{\mathbb{R}^N\times\mathbb{R}},\;\tau\in [0,1],
\end{equation*}
where $\mathcal{F}(x,t):=\frac{1}{p}f(x,t)t-F(x,t)$ and
$F(x,t):=\int^t_0 f(x,\tau) d\tau$;

\item[(A5)] $F(x,0)\equiv0, F(x,t)\geq0$ for all 
$(x,t)\in \mathbb{R}^N\times\mathbb{R}$ and
\begin{equation*}
  \lim_{|t|\to  +\infty}\frac{F(x,t)}{|t|^p}=+\infty\quad
\text{uniformly in }x\in \mathbb{R}^N;.
\end{equation*}

\item[(A6)] $f(x,-t)=-f(x,t)$, for all  $(x,t)\in{\mathbb{R}^N\times{\mathbb{R}}}$.
\end{itemize}

 Specifically, the author established the following theorem in \cite{VA}.

 \begin{theorem}[{\cite[Theorem 1]{VA}}] \label{thm1.1}
 Assume that {\rm (A1)--(A6)} are satisfied.
Then  \eqref{q}   has infinitely many nontrivial weak solutions.
\end{theorem}


We note that the usual condition $f(x,u)/u\to 0$ as $u\to 0$ is not needed in
Theorem \ref{thm1.1}. This is the highlights in \cite{VA}. Conditions like (A1) and (A2)
have been given by Bartsch, Wang and Willem \cite{T4}. Condition (A4)
is due to Jeanjean \cite{LJ}. This condition is also used together with
a Cerami type argument in singularly perturbed elliptic problems in $\mathbb{R}^N$
with autonomous nonlinearity. Moreover, there are many functions (e.g.
$f(x,t)=a|t|^{p-2}t\ln(1+|t|), a>0)$ which satisfy (A4), but do not satisfy
the following classical condition:
\begin{itemize}
\item[(AR)] there exists $\mu>p$ such that
$$
0<\mu F(x,t)\leq tf(x,t),\quad t\neq0.
$$
\end{itemize}
However,  condition (AR) does not imply condition (A4); see the example
in \cite{T11CC}.

Motivated by the above works, we shall further study the infinitely many
nontrivial solutions of   \eqref{q} with sign-changing potential and subcritical
$p$-superlinear nonlinearity. Moreover, we are interested in the case where
the potential $V$ and the primitive of $f$ are both sign-changing, which is
called a double sign-changing case and prevents us from applying a
standard variational argument directly. For the above reasons,
just a few papers dealt with such a double sign-changing case as regards
 \eqref{q} until now. We will give a direct sum decomposition of the
fractional Sobolev space and establish some new theorems on the infinitely
many nontrivial solutions of   \eqref{q}
 with mild assumptions deeply different from those studied in previous related
works. For any $\eta>0$, the Sobolev embedding theorem implies
$E^s(B_\eta) \hookrightarrow {L^2(B_\eta)}$, where
$B_\eta = \{x\in{\mathbb{R}^N}: |x|<\eta\}$. Based on the above fact, we
can construct a direct sum decomposition of $E^s(B_\eta)$.
As far as we know, there were no such multiplicity results in this situation.

To state our results, we introduce the following assumptions:
\begin{itemize}
\item[(A5')] $\lim_{|t|\to +\infty}\frac{F(x,t)}{|t|^p}=+\infty$, a.e.
$x\in{\mathbb{R}^N}$, and there exists $r_0\geq0$ such that
\begin{equation*}
  F(x,t)\geq0,\quad \forall(x,t)\in{{\mathbb{R}^N}\times\mathbb{R}},\; |t|\geq r_0;
\end{equation*}

\item[(A7)] $\mathcal{F}(x,t):=\frac{1}{p}f(x,t)t-F(x,t)\geq0$, and there exist
$c_0>0$ and $\kappa>{\max\{1,\frac{N}{ps}}\}$ such that
\begin{equation*}
  |F(x,t)|^\kappa \leq c_0|t|^{p\kappa}\mathcal{F}(x,t),\quad
\forall (x,t)\in {\mathbb{R}^N\times\mathbb{R}},\; |t|\geq r_0;
\end{equation*}

\item[(A8)] there exist $\mu>p$ and $\varrho>0$ such that
\begin{equation*}
  \mu F(x,t)\leq tf(x,t)+\varrho |t|^p,\quad \forall (x,t)
\in{\mathbb{R}^N\times\mathbb{R}};
\end{equation*}

\item[(A9)] there exist $\mu>p$ and $r_1>0$ such that
\begin{equation*}
  \mu F(x,t)\leq tf(x,t),\quad \forall (x,t)\in{\mathbb{R}^N\times\mathbb{R}},\;
 |t|\geq r_1;
\end{equation*}

\item[(A10)] $f(x,t)=o(|t|^{p-1})$, as $|t|\to 0$, uniformly 
in $x\in \mathbb{R}^N$.

\end{itemize}
It is easy to check that (A3) and (A9) imply (A8). Now we are ready
to state the main results of this paper.

\begin{theorem} \label{thm1.2}
 Assume that {\rm (A1)--(A3), (A5'), (A6), (A7)} are satisfied.
Then  \eqref{q} possesses infinitely many nontrivial solutions
\end{theorem}

\begin{theorem} \label{thm1.3}
 Assume that {\rm (A1)--(A3), (A5'), (A6), (A8)} are satisfied.
Then  \eqref{q} possesses infinitely many nontrivial solutions.
\end{theorem}

\begin{corollary} \label{coro1.4}
 Assume that {\rm (A1)-(A3), (A5'), (A6), (A9)} are satisfied.
Then  \eqref{q} possesses infinitely many nontrivial solutions.
\end{corollary}

\begin{remark} \label{rmk1.5} \rm
It is easy to see that (A5') and (A7)  are weaker than (A5) and (AR), respectively.
In particular, we remove the
usual condition (A10), and $F(x,t)$ is allowed to be sign-changing in
Theorems \ref{thm1.2}, \ref{thm1.3} and Corollary \ref{coro1.4}. The role of (AR) is to ensure the
 boundedness of the Palais-Smale (PS) sequences of the energy functional,
it is also significant to construct the variational framework.
This is very crucial in applying the critical point theory.
However, there are many functions which are superlinear at infinity,
but do not satisfy the condition (AR) for any $\mu>p$. For example,
the superlinear function $f(x,t)$ for the case $p=2$:
\begin{equation}\label{88}
  f(x,t)=a(x)t\ln(1+|t|),
\end{equation}
where $0<\inf_{x\in \mathbb{R}^N}a(x)\leq\sup_{x\in \mathbb{R}^N}a(x)<+\infty$.
Indeed, (AR) implies that $f(x,u)\geq C|u|^\mu$ for some $C>0$, and so
\eqref{88} not satisfies (AR). It is easy to check function
$f(x,t)=b|t|^{p-2}t\ln(1+|t|)$ with $b>0$ satisfies (A3), (A4), (A5), (A7) and (A10).
 However, the  function
\begin{equation}\label{78}
 f(x,t)=3t|t|\int^t_0|\tau|^{1+\sin\tau}\tau d\tau+|t|^{4+\sin t}t,
\end{equation}
satisfies (A7) not (A4) for $p=2$, see \cite[Section 3]{T11CC}.
In addition, the  function
\begin{equation}\label{99}
  f(x,t)=d(x)|t|^{p-1}t[(p+3)t^2-2(p+2)t+p+1]
\end{equation}
satisfies (A3), (A5'), (A6), (A9) and (A10), where
\[
0<\inf_{x\in\mathbb{R}^N}d(x)\leq \sup_{x\in\mathbb{R}^N}d(x)<+\infty.
\]
One can see that \eqref{99} satisfies neither (AR) nor (A4), see \cite[Section 1]{13Sh}.
\end{remark}

On the existence of ground state solutions we have the following result.

\begin{theorem} \label{thm1.6}
 Assume {\rm (A1)--(A3), (A5'), (A6), (A7)} and {\rm (A10)} are satisfied.
Then  \eqref{q}  has a ground state solution $u_0$  such that
 $J(u_0)=\inf_{u\in\mathcal{M}}J(u)$,  where $\mathcal{M}=\{u\neq0,J'(u)=0\}$.
\end{theorem}

\section{Variational setting and proofs the main results}

Throughout this section, we make the following assumption instead of (A1):
\begin{itemize}
\item[(A1')] $V\in C(\mathbb{R}^N,\mathbb{R})$ and $\inf_{x\in\mathbb{R}^N}V(x)>0$.
\end{itemize}
Firstly, we give some notation related to the fractional Sobolev space
$W^{s,p}(\mathbb{R}^N)$. For $0<s<1$ and $p\geq2$, define the so-called
Gagliardo seminorm by
$$
[u]_{s,p}:=\Big(\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)
|^p}{|x-y|^{N+ps}}\,dx\,dy\Big)^{1/p},
$$
where $u : \mathbb{R}^N\mapsto\mathbb{R}$ is a measurable function.
Then the fraction Sobolev space $W^{s,p}(\mathbb{R}^N)$ is given by
$$
W^{s,p}(\mathbb{R}^N):=\{u\in L^p(\mathbb{R}^N): u\text{ is measurable }[u]<\infty\},
$$
which can be equipped with the norm
$$
\|u\|_{W^{s,p}(\mathbb{R}^N)}=\Big([u]^p_{s,p}+\|u\|^p_{L^p}\Big)^{1/p},
$$
where $\|u\|^p_{L^p}=\int_{\mathbb{R}^N}|u(x)|^p dx$.
By condition (A1'), we define the fractional Sobolev space with potential
$V(x)$ by
$$
E^s:=\big\{u\in W^{s,p}(\mathbb{R}^N) :
\int_{\mathbb{R}^N}V(x)|u|^p dx<\infty\big\}
$$
equipped with the norm
$$
\|u\|_{E^s}=([u]^p_{s,p}+\|V^\frac{1}{p}u\|^p_{L^p})^\frac{1}{p}.
$$

\begin{lemma}[{\cite[Theorem 6.5]{DI7}}] \label{lem2.1}
 Under  assumption {\rm (A1')}, for any $r\in[p,p^*_s]$,  the embedding
\begin{equation*}
E^s\hookrightarrow W^{s,p}(\mathbb{R}^N)\hookrightarrow L^r(\mathbb{R}^N)
\end{equation*}
 is continuous.
 In particular, there exist constants $\rho_r>0$ and $\gamma_r>0$ such that
\begin{equation}\label{p0}
  \|u\|_{L^r}\leq \rho_r\|u\|_{W^{s,p}}
\leq\gamma_r\|u\|_{E^s},\quad \forall  u\in E^s.
\end{equation}
\end{lemma}

\begin{lemma}[{\cite[Lemma 1]{VA}}]  \label{lem2.2}
 Under assumptions {\rm (A1')} and {\rm (A2)},  the embedding
$E^s\hookrightarrow L^r(\mathbb{R}^N)$  is compact for any
$r\in [p,p^*_s)$.
\end{lemma}

Now we define a functional $J$ on $E^s$ by
\begin{equation}\label{p}
\begin{aligned}
&J(u)\\
&=\frac{1}{p}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
 \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}} \,dx\,dy
 +\frac{1}{p}\int_{\mathbb{R}^N}V(x){|u|^p} dx
 -\int_{\mathbb{R}^N}F(x,u) dx  \\
&=\frac{1}{p}\|u\|^p_{E^s}-\Phi(u),
\end{aligned}
\end{equation}
where $\Phi(u)=\int_{\mathbb{R}^N}F(x,u) dx$, for all $u\in E^s$.
We say that $u\in E^s$ is a weak solution for  \eqref{q} if
\begin{align*}
&\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))
 (h(x)-h(y))}{|x-y|^{N+ps}} \,dx\,dy\\
&+\int_{\mathbb{R}^N}V(x){|u|^{p-2}uh} dx
-\int_{\mathbb{R}^N}f(x,u)h dx=0, \quad
\forall h\in E^s.
\end{align*}
Note that critical points of $J$ correspond to weak solutions of
equation \eqref{q}.

\begin{lemma}[\cite{CHEN}] \label{lem2.3}
 Under assumptions {\rm (A1'), (A3)}. The
functional  $\Phi\in C^1(E^s,\mathbb{R})$  and
 $ \langle\Phi'(u),h\rangle=\int_{\mathbb{R}^N}f(x,u)h dx$  for all
 $u,h\in E^s$.  Moreover $\Phi':E^s\mapsto {E^s}^*$ is weakly continuous.
\end{lemma}

The proof of the above lemma is similar to the proof in \cite{CHEN},
we omit the proof.

From the above facts, we know $J$ is well defined in $E^s$ and
$J\in C^1(E^s,\mathbb{R})$. Moreover,
\begin{equation}\label{p4}
\begin{aligned}
  \langle J'(u),h\rangle
&=\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)
 -u(y))(h(x)-h(y))}{|x-y|^{N+ps}}\,dx\,dy\\
&\quad +\int_{\mathbb{R}^N}V(x){|u|^{p-2}uh} dx-\langle\Phi'(u),h\rangle
  \end{aligned}
\end{equation}
for all $u,h\in E^s$. Obviously, solutions for equation \eqref{q} are
correspond to critical points of the energy functional $J$.

Define the nonlinear operator $\Upsilon$: $E^s\times E^s \mapsto \mathbb{R}$ by
\begin{equation*}
  \Upsilon(u,h)
=\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))
(h(x)-h(y))}{|x-y|^{N+ps}}\,dx\,dy, \quad \forall  u,h\in E^s.
\end{equation*}
We say that $I\in C^1(X,\mathbb{R})$ satisfies $(C)_c$-condition if any sequence
$\{u_n\}\subset X$ such that
\begin{equation}\label{p5}
  I(u_n)\to  c, \quad \|I'(u_n)\|(1+\|u_n\|)\to  0
\end{equation}
has a convergent subsequence.

\begin{lemma}[\cite{T11CC,T11T}] \label{lem2.4}
 Let  $X$  be an infinite dimensional Banach space,
$X=Y\oplus Z$,  where $Y$  is finite dimensional.
 If $I\in C^1(X,\mathbb{R})$  satisfies the $(C)_c$-condition for all $c>0$,  and
\begin{itemize}
\item[(1)] $I(0)=0$, $I(-u)=I(u)$  for all  $u\in X$;
\item[(2)] there exist constants $\rho$,~$\alpha>0$  such that
  $I|_{\partial B\rho\cap Z}$ $\geq\alpha$;
\item[(3)]  for any finite dimensional subspace
 $\widetilde X\subset X$, there is $R=R(\widetilde X)>0$  such that
 $I(u)\leq0$  on  $\widetilde X\backslash B_R$.  Then $I$
 possesses an unbounded sequence of critical values.
\end{itemize}
\end{lemma}


\begin{lemma} \label{lem2.5}
 Under assumptions {\rm (A1'), (A2), (A3), (A5'), (A7)}, any sequence of
$ \{u_n\}\subset E^s$  satisfying
\begin{equation}\label{p6}
{J(u_n)}\to  {c>0}, \quad \langle J'(u_n),u_n\rangle \to 0
\end{equation}
is bounded in $E^s$.
\end{lemma}

\begin{proof}
To achieve our goals, arguing by contradiction, suppose that
 $\|u_n\|_{E^s}\to \infty$. Let $v_n=\frac{u_n}{\|u_n\|_{E^s}}$,
then $\|v_n\|_{E^s}=1$ and $\|v_n\|_{L^r}\leq\gamma_r\|v_n\|_{E^s}=\gamma_r$
for $p\leq r<p^*_s$. Observe that for $n$ large
\begin{equation}\label{p7}
  c+1\geq J(u_n)-\frac{1}{p}\langle J'(u_n),u_n \rangle
=\int_{\mathbb{R}^N}\mathcal{F}(x,u_n) dx,
\end{equation}
where $\mathcal{F}(x,u_n)=\frac{1}{p}f(x,u_n)u_n-F(x,u_n)\geq0$.
By condition (A3), we obtain
 \begin{equation}\label{p8}
   |F(x,t)|\leq\frac{c_1}{p}|t|^p+\frac{c_2}{q}|t|^q.
 \end{equation}
For $0\leq a <b$, let
 \begin{equation}\label{p9}
   \Omega_n(a,b)=\{x\in {\mathbb{R}^N}:a\leq |u_n(x)|<b\}.
 \end{equation}
 Going if necessary to a subsequence, we may assume that
 \begin{equation}\label{p10}
 \begin{gathered}
   v_n\rightharpoonup v\quad \text{in }E^s\\
   v_n\to  v \quad \text{in } L^r(\mathbb{R}^N),\; p\leq r<p^*_s\\
   v_n(x)\to  v(x) \quad \text{a.e. on } \mathbb{R}^N.
   \end{gathered}
 \end{equation}
 Now, we consider two possible cases: $v=0$ or $v\neq0$.
\smallskip

\noindent\textbf{Case 1:} if $v=0$, then $v_n\to  v$  in
$L^r(\mathbb{R}^N)$, $p\leq r<p^*_s$, and $v_n\to  0$  a.e. on
$\mathbb{R}^N$. Note that
\begin{equation*}
  J(u_n)=\frac{1}{p}\|u_n\|^p_{E^s}-\int_{\mathbb{R}^N}F(x,u_n) dx.
\end{equation*}
This, together with \eqref{p6}, implies that
\begin{equation}\label{p11}
  \limsup_{n\to \infty}\int_{\mathbb{R}^N}\frac{|F(x,u_n)|}{\|u_n\|^p_{E^s}} dx
\geq\frac{1}{p}.
\end{equation}
 On the other hand, by \eqref{p8}, one has
\begin{equation}\label{p12}
 \begin{aligned}
  \int_{\Omega_n(0,r_0)}\frac{|F(x,u_n)|}{\|u_n\|^p_{E^s}} dx
&=\int_{\Omega_n(0,r_0)}\frac{|F(x,u_n)|}{|u_n|^p}\frac{|u_n|^p}{\|u_n\|^p_{E^s}} dx\\
&=\int_{\Omega_n(0,r_0)}\frac{|F(x,u_n)|}{|u_n|^p}|v_n|^p dx\\
&\leq\Big(\frac{c_1}{p}r_0+\frac{c_2}{q}{r_0}^{q-p}\Big)
 \int_{\Omega_n(0,r_0)}|v_n|^p dx\\
&\leq\Big(\frac{c_1}{p}r_0+\frac{c_2}{q}{r_0}^{q-p}\Big)\|v_n\|^p_{L^p}\to 0,
\quad \text{as } n\to \infty.
  \end{aligned}
\end{equation}
Set $\kappa'=\kappa/(\kappa-1)$, $\kappa>\max\{1,N/ps\}$, then
${p\kappa'}$ $\in [p,p^*_s)$. Hence, from (A7), \eqref{p7} and \eqref{p10}, one has
\begin{equation}\label{p13}
 \begin{aligned}
&\int_{\Omega_n(r_0,+\infty)}\frac{|F(x,u_n)|}{\|u_n\|^p_{E^s}} dx \\
&=\int_{\Omega_n(r_0,+\infty)}\frac{|F(x,u_n)|}{|u_n|^p}|v_n|^p dx\\
&\leq\Big[\int_{\Omega_n(r_0,+\infty)}
 \Big(\frac{|F(x,u_n)|}{|u_n|^p}\Big)^{\kappa} dx\Big]^{1/\kappa}
\Big[\int_{\Omega_n(r_0,+\infty)}|v_n|^{p\kappa'} dx\Big]^{1/\kappa'}\\
&\leq{c_0}^{1/\kappa}\Big[\int_{\Omega_n(r_0,+\infty)}\mathcal{F}(x,u_n) dx
 \Big]^{1/\kappa}\Big[\int_{\Omega_n(r_0,+\infty)}|v_n|^{p\kappa'} dx\Big]^{1/\kappa'}\\
&\leq\left[c_0(c+1)\right]^{1/\kappa}
\Big[\int_{\Omega_n(r_0,+\infty)}|v_n|^{p\kappa'} dx\Big]^{1/\kappa'}\\
&\leq\left[c_0(c+1)\right]^{1/\kappa}\|v_n\|^p_{L^{p\kappa'}}\to 0,
\quad \text{as } n\to \infty.
  \end{aligned}
\end{equation}
Combining \eqref{p12} with \eqref{p13}, we have
\begin{align*}
&\int_{\mathbb{R}^N}\frac{|F(x,u_n)|}{\|u_n\|^p_{E^s}} dx \\
&=\int_{\Omega_n(0,r_0)}\frac{|F(x,u_n)|}{|u_n|^p}|v_n|^pdx
+\int_{\Omega_n(r_0,+\infty)}\frac{|F(x,u_n)|}{|u_n|^p}|v_n|^p dx\to 0,
\quad \text{as } n\to \infty,
\end{align*}
which contradicts \eqref{p11}.
\smallskip

\noindent\textbf{Case 2:} $v\neq0$.
Set $A:=\{x\in {\mathbb{R}^N}: v(x)\neq0\}$, then meas$(A)>0$.
For a.e. $x\in A$, we have $\lim_{n\to \infty}|u_n(x)|=+\infty$.
Hence $A\subset\Omega_n(r_0,+\infty)$ for large $n\in \mathbb{N}$.
It follows from (A3), (A5'), \eqref{p10} and Fatou's Lemma that
\begin{equation}\label{p14}
\begin{aligned}
  0&=\lim_{n\to \infty}\frac{c+o(1)}{\|u_n\|^p_{E^s}}
 =\lim_{n\to \infty}\frac{J(u_n)}{\|u_n\|^p_{E^s}}\\
&=\lim_{n\to \infty}\Big[\frac{1}{p}-\int_{\mathbb{R}^N}
 \frac{F(x,u_n)}{\|u_n\|^p_{E^s}}\Big]\\
&\leq\lim_{n\to \infty}\Big[\frac{1}{p}+\int_{\Omega_n(0,r_0)}
\frac{|F(x,u_n)|}{|u_n|^p}|v_n|^p dx \\
&\quad -\int_{\Omega_n(r_0,+\infty)}\frac{|F(x,u_n)|}{|u_n|^p}|v_n|^p dx\Big]\\
&\leq\frac{1}{p}+\limsup_{n\to \infty}\int_{\Omega_n(0,r_0)}
 (\frac{c_1}{p}+\frac{c_2}{q}|u_n|^{q-p})|v_n|^p dx \\
&\quad  -\liminf_{n\to \infty}\int_{\Omega_n(r_0,+\infty)}
 \frac{|F(x,u_n)|}{|u_n|^p}|v_n|^p dx\\
&\leq\frac{1}{p}+(\frac{c_1}{p}+\frac{c_2}{q}{r_0}^{q-p})\gamma^p_p\\
&\quad -\liminf_{n\to \infty}\int_{\mathbb{R}^N}\frac{|F(x,u_n)|}{|u_n|^p}
 [\chi_{\Omega_n(r_0,+\infty)}(x)]|v_n|^p dx\\
&\leq\frac{1}{p}+(\frac{c_1}{p}+\frac{c_2}{q}{r_0}^{q-p})\gamma^p_p\\
&\quad -\int_{\mathbb{R}^N}\liminf_{n\to \infty}\frac{|F(x,u_n)|}
 {|u_n|^p}[\chi_{\Omega_n(r_0,+\infty)}(x)]|v_n|^p dx
=-\infty,
 \end{aligned}
\end{equation}
which is a contradiction. Thus $\{u_n\}$ is bounded in $E^s$.
\end{proof}

\begin{lemma} \label{lem2.6}
Under assumptions {\rm (A1'), (A2), (A3), (A5'), (A7)},  any $(C)_c$-sequence
of $J$ has a convergent subsequence in $E^s$.
\end{lemma}

\begin{proof}
 Let $\{u_n\}$ be a $(C)_c$-sequence of $J$, then
 \begin{equation}\label{p15'}
  J(u_n)\to  c, \quad
\|J'(u_n)\|(1+\|u_n\|_{E^s})\to 0 ,\quad
\sup_{n\in\mathbb{N}}\|u_n\|_{E^s}<+\infty.
\end{equation}
Lemma \ref{lem2.5} implies that $\{u_n\}$ is bounded in $E^s$. Going if necessary
 to a subsequence, we can assume that $u_n\rightharpoonup u$ in $E^s$.
By Lemma \ref{lem2.2}, $u_n\to  u$ in $L^r(\mathbb{R}^N)$ for $p\leq r<p^*_s$.
By a calculation, it follows from \eqref{p15'} that
\begin{equation}\label{p16'}
\begin{aligned}
  o(1)
&=\langle J'(u_n)-J'(u),u_n-u\rangle\\
&=\Upsilon(u_n,u_n-u)-\Upsilon(u,u_n-u)\\
&\quad   +\int_{\mathbb{R}^N}V(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)(u_n-u) dx\\
&\quad -\langle\Phi'(u_n)-\Phi'(u),u_n-u\rangle.
\end{aligned}
\end{equation}
Firstly, to prove our results, we need recall the well-known Simion inequality
\begin{equation}\label{0}
  (|a|^{p-2}a-|b|^{p-2}b)(a-b)\geq k_p|a-b|^p,\quad k_p>0,\quad
\forall {a,\,b}\in\mathbb{R}
\end{equation}
for $p\geq2$.
By \eqref{0}, we have
\begin{gather}\label{p17'}
  \Upsilon(u_n,u_n-u)-\Upsilon(u,u_n-u)\geq k_p\Upsilon(u_n-u,u_n-u)
=k_p[u_n-u]^p_{s,p}, \\
\label{p18'}
 \int_{\mathbb{R}^N}V(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)(u_n-u) dx
\geq k_p\int_{\mathbb{R}^N}V(x)|u_n-u|^p dx.
\end{gather}
Secondly, in view of (A3), by the H\"{o}lder inequality, we have
\begin{align*}
 &\Big|\int_{\mathbb{R}^N}\left[f(x,u_n)-f(x,u)\right](u_n-u) dx\Big|\\
&\leq(c_1+c_2)\int_{\mathbb{R}^N}\left[|u_n|^{p-1}+|u|^{p-1}+|u_n|^{q-1}
 +|u|^{q-1}\right]|u_n-u| dx\\
&\leq(c_1+c_2)[(\|u_n\|^{p-1}_{L^p}+\|u\|^{p-1}_{L^p})\|u_n-u\|_{L^p} \\
&\quad  +(\|u_n\|^{q-1}_{L^q}+\|u\|^{q-1}_{L^q})\|u_n-u\|_{L^q}],
  \end{align*}
which implies that
\begin{equation}\label{p19'}
  \langle\Phi'(u_n)-\Phi'(u),u_n-u\rangle\to 0,\quad \text{as }n\to \infty.
\end{equation}
Finally, the combination of \eqref{p16'}-\eqref{p19'} implies
\begin{equation}\label{p100}
  o(1)\geq k_p\|u_n-u\|^p_{E^s}+o(1).
\end{equation}
Then
\[
  u_n\to  u\quad \text{in }E^s,\quad \text{as } n\to \infty.
\]
This completes the proof.
\end{proof}

\begin{lemma} \label{lem2.7}
Under assumptions {\rm (A1'), (A2), (A3), (A5'), (A8)}, every $(C)_c$-se\-quence
 of $J$  has a convergent subsequence in $E^s$.
\end{lemma}

\begin{proof}
 Employing Lemma \ref{lem2.6}, we only prove that $\{u_n\}$ is bounded in $E^s$.
Arguing by contradiction, we suppose that $\|u_n\|_{E^s}\to \infty$.
Let $v_n=\frac{u_n}{\|u_n\|_{E^s}}$, then $\|v_n\|_{E^s}=1$ and
$\|v_n\|_{L^r}\leq \gamma_r\|v_n\|_{E^s}=\gamma_r$ for $p\leq r<p^*_s$.
Passing to a subsequence, we may assume that $v_n\rightharpoonup v$
in $E^s$, by Lemma \ref{lem2.2}, $v_n\to  v$ in $L^r(\mathbb{R}^N)$, $p\leq r<p^*_s$,
and $v_n\to  v$ a.e. on $\mathbb{R}^N$.
By \eqref{p}, \eqref{p4}, \eqref{p5} and (A8), one has
\begin{align*}
c+1&\geq J(u_n)-\frac{1}{\mu}\langle J'(u_n),u_n\rangle\\
&=\frac{\mu-p}{p\mu}\|u_n\|^p_{E^s}+\int_{\mathbb{R}^N}
\Big[\frac{1}{\mu}f(x,u_n)u-F(x,u_n)\Big] dx\\
&\geq\frac{\mu-p}{p\mu}\|u_n\|^p_{E^s}-\frac{\varrho}{\mu}\|u_n\|^p_{L^p},
   \end{align*}
for large $n\in\mathbb{N}$, which implies
\begin{equation}\label{p25}
  1\leq\frac{p\varrho}{\mu-p}\limsup_{n\to \infty}\|v_n\|^p_{L^p}.
\end{equation}
Hence, it follows from \eqref{p25} that $v\neq0$.
By a similar process as in \eqref{p14}, we can conclude a contradiction.
Thus, $\{u_n\}$ is bounded in $E^s$. The rest proof is the same as that
 in Lemma \ref{lem2.6}.
\end{proof}

\begin{lemma} \label{lem2.8}
 Under assumptions {\rm (A1'), (A2), (A3), (A5')}, for any finite dimensional
subspace $\widetilde E^s\subset E^s$,  we have
\begin{equation}\label{p26}
  J(u)\to -\infty,\quad \text{as } \|u\|_{E^s}\to \infty,\quad
u\in \widetilde E^s.
\end{equation}
\end{lemma}

\begin{proof}
 Arguing indirectly, assume that for some sequence $\{u_n\}\subset\widetilde E^s$
with $\|u_n\|_{E^s}\to \infty$, there is $M>0$ such that $J(u_n)\geq -M$ for all
$n\in \mathbb{N}$. Set $v_n=\frac{u_n}{\|u_n\|_{E^s}}$, then
$\|v_n\|_{E^s}=1$. Passing to a subsequence, we may assume that
 $v_n\rightharpoonup v\in\widetilde{E}^s$. Since $\widetilde E^s$
is finite dimensional, then $v_n\to  v\in\widetilde E^s$ in $E^s$, $v_n\to  v$
a.e. on $\mathbb{R}^N$, and so $\|v\|_{E^s}=1$. Hence, we can deduce a
contradiction in a similar way as \eqref{p14}.
\end{proof}

\begin{corollary} \label{coro2.9}
Under assumptions {\rm (A1'), (A2), (A3), (A5')}, for any finite dimensional
 subspace $\widetilde E^s\subset E^s$, there is $R=R(\widetilde E^s)>0$
 such that
\begin{equation}\label{p27}
  J(u)\leq0,\quad \forall u\in \widetilde E^s,\quad \|u\|_{E^s}\geq R.
\end{equation}
\end{corollary}

\begin{lemma} \label{lem2.10}
 Let $\Omega:=\{(x_1,x_2,\dots,x_N)\in {\mathbb{R}^N}:
 0\leq x_i<\frac{d_0}{\sqrt{N}},i=1,2,\dots,N\}$.
Then there exists a constant $a_0\geq0$  such that
\begin{equation}\label{p28}
  \Big(\int_{\Omega}|u|^{p\lambda} dx\Big)^{1/\lambda}
\leq a_0\|u\|^p_{E^s},
\end{equation}
where $\lambda=2$ if $sp< N$ and $\lambda={(2N-sp)/(2N-2sp)}$ if $sp\leq N$.
\end{lemma}

The proof of the above lemma, follows from Lemma \ref{lem2.1};
it is easy to see that \eqref{p28} holds.

\begin{lemma} \label{lem2.11}
Under assumptions {\rm (A1')} and {\rm (A2)}, for any $r>0$  and $M>0$,
\begin{equation}\label{p29}
\int_{|x|>r}|u|^p dx
\leq\left\{\frac{1}{M}+a_0[\varepsilon_r(M)]^{(\lambda-1)/\lambda}\right\}
\|u\|^p_{E^s},\quad \forall u\in {E^s},
\end{equation}
where
\begin{equation*}
  \varepsilon_r(M):=\sup_{|y|\geq r}\operatorname{meas}\{x\in {\mathbb{R}^N}:
|x-y|\leq d_0, V(x)\leq M\}.
\end{equation*}
\end{lemma}

\begin{proof}
 Let $\Omega:=\{(x_1,x_2,\dots,x_N)\in {\mathbb{R}^N}:
0\leq x_i<\frac{d_0}{\sqrt{N}},i=1,2,\dots,N\}$.
Then $\Omega\subset B_{d_0}$. Take a sequence
$\{z^i\}_{i\in \mathbb{N}}\subset {\mathbb{R}^N}$ such that
\begin{equation}\label{p30}
  \mathbb{R}^N=\cup_{i\in\mathbb{N}}(z^i+\Omega),\quad {(z^i+\Omega)
\cap(z^j+\Omega)=\emptyset},\quad \forall i\neq j.
\end{equation}
Let $\Lambda_i=z^i+\Omega$. For any $r>0$ and $M>0$, let
\begin{gather*}
  A(r,M):=\{x\in{\mathbb{R}^N}: |x|>r, V(x)>M\}, \\
B(r,M):=\{x\in{\mathbb{R}^N}: |x|>r, V(x)\leq M\}.
\end{gather*}
Then
\begin{equation}\label{p31}
\begin{aligned}
  \int_{A(r,M)}|u|^p dx
&\leq \frac{1}{M}\int_{A(r,M)}V(x)|u|^p dx \\
&\leq\frac{1}{M}\int_{\mathbb{R}^N}V(x)|u|^p dx
\leq\frac{1}{M}\|u\|^p_{E^s}.
\end{aligned}
\end{equation}
On the other hand, from \eqref{p28}, \eqref{p30} and the H\"{o}lder inequality,
one has
\begin{equation}\label{p32}
  \begin{aligned}
  \int_{B(r,M)}|u|^p dx
&=\sum_{i\in \mathbb{N}}\int_{B(r,M)\cap{\Lambda_i}}|u|^p dx\\
&\leq\sum_{i\in \mathbb{N}}\left[\operatorname{meas}({B(r,M)\cap{\Lambda_i}})
 \right]^{(\lambda-1)/\lambda}
\Big(\int_{B(r,M)\cap{\Lambda_i}}|u|^{p\lambda} dx\Big)^{1/\lambda}\\
&\leq[\varepsilon_r(M)]^{(\lambda-1)/\lambda}\sum_{i\in \mathbb{N}}
\Big(\int_{B(r,M)\cap{\Lambda_i}}|u|^{p\lambda} dx\Big)^{1/\lambda}\\
&\leq a_0[\varepsilon_r(M)]^{(\lambda-1)/\lambda}\|u\|^p_{E^s}.
   \end{aligned}
\end{equation}
Both \eqref{p30} and \eqref{p31} imply that \eqref{p29} holds.
\end{proof}

Since $\varepsilon_r(M)\to 0$ as $r\to \infty$, by Lemma \ref{lem2.11}, we can choose
$\eta_0>0$ such that
\begin{equation}\label{p33}
  \int_{|x|>{\eta_0}}|u|^p dx\leq\frac{1}{2pc_1}\|u\|^p_{E^s},
\quad \forall u\in {E^s}.
\end{equation}
Let $\{e_j\}$ be a total orthonormal basis of $L^2(B_{\eta_0})$ and define
 $X_j=\mathbb{R}e_j,j\in\mathbb{N}$,
\begin{equation}\label{p35}
  Y_k=\oplus^k_{j=1}X_j,\quad ~Z_k=\oplus^\infty_{j=k+1}X_j,\quad
K\in\mathbb{N}.
\end{equation}

\begin{lemma} \label{lem2.12}
Under assumptions {\rm (A1')}  and {\rm (A2)}, for $p\leq r<p^*_s$,
\begin{equation}\label{p36}
  \beta^s_k:=\sup_{u\in Z_k,\|u\|_{E^s(B_{\eta_o})}=1}\|u\|_{L^r(B_{\eta_0})}
\to 0,\quad\text{as }k\to \infty.
\end{equation}
\end{lemma}

\begin{proof}
 Note that $E^s(B_{\eta_0})\hookrightarrow L^r(B_{\eta_0})$ for
$1\leq r<p^*_s$. It is clear that $0<\beta^s_{k+1}$ $\leq{\beta^s_k}$,
and so that $\beta^s_k\to  \beta^s\geq0,k\to \infty$.
For every $k\geq0$, there exists $u_k\in Z_k$ such that
$\|u_k\|_{E^s(B_{\eta_0})}=1$ and $\|u_k\|_{L^r(B_{\eta_0})}>{\beta^s_k}/2$.
 By definition of $Z_k$, $u_k\rightharpoonup0$ in $L^2(B_{{\eta_0}})$, and so
$u_k\rightharpoonup0$ in $E^s(B_{{\eta_0}})$.
Lemma \ref{lem2.2} implies that $u_k\to 0$ in $L^r(B_{{\eta_0}})$.
Thus we have proved that $\beta^s=0$.
\end{proof}

By Lemmas \ref{lem2.1} and \ref{lem2.12}, for all $u\in {Z_m}\cap{W^{s,p}(B_{\eta_0})}$,
we can take an integer $m\geq1$ such that
\begin{equation}\label{p37}
 \begin{aligned}
&\int_{|x|\leq {\eta_0}}|u|^p dx \\
&\leq\frac{1}{2 pc_1}\Big[\int_{|x|\leq{\eta_0}}\int_{|x|\leq{\eta_0}}
\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}} \,dx\,dy+\int_{|x|\leq{\eta_0}}
V(x)|u|^p dx\Big]
\end{aligned}
\end{equation}
Let $\zeta(x)=0$ if $|x|\leq\eta_0$ and $\zeta(x)=1$ if $|x|>\eta_0$. Define
\begin{gather}\label{p38}
  Y=\{(1-\zeta)u: u\in E^s, (1-\zeta)u\in Y_m\}, \\
\label{p39}
  Z=\{(1-\zeta)u: u\in E^s, (1-\zeta)u\in Z_m\}+\{\zeta v: v\in E^s\}.
\end{gather}
Then $Y$ and $Z$ are subspaces of $E^s$, and $E^s=Y\oplus Z$.

\begin{lemma} \label{lem2.13}
 Under assumptions {\rm  (A1'), (A2), (A3)}, there exist constants
$\rho,\alpha>0$ such that $J|_{\partial{B_\rho}\cap Z}\geq\alpha$.
\end{lemma}

\begin{proof}
 By \eqref{p33}, \eqref{p37} and \eqref{p39}, we have
\begin{equation}\label{t3}
\begin{aligned}
  \|u\|^p_{L^p}
&=\int_{|x|\leq {\eta_0}}|u|^p dx+\int_{|x|>{\eta_0}}|u|^p dx\\
  &\leq\frac{1}{2pc_1}\Big[\int_{|x|\leq{\eta_0}}\int_{|x|\leq{\eta_0}}\frac{|u(x)
 -u(y)|^p}{|x-y|^{N+ps}} \,dx\,dy \\
&\quad +\int_{|x|\leq{\eta_0}}|V(x)^{1/p}u|^p dx\Big]
 +\frac{1}{2pc_1}\|u\|^p_{E^s}\\
&\leq\frac{1}{2pc_1}\|u\|^p_{E^s}+\frac{1}{2pc_1}\|u\|^p_{E^s}\\
&\leq\frac{1}{pc_1}\|u\|^p_{E^s},\quad \forall  u\in Z.
  \end{aligned}
\end{equation}
Hence, it follows from (A3), \eqref{p0}, \eqref{p} and \eqref{t3} that
\begin{align*}
 J(u)&=\frac{1}{p}\|u\|^p_{E^s}-\Phi(u)\\
 &\geq\frac{1}{p}\|u\|^p_{E^s}-\frac{c_1}{p}\|u\|^p_{L^p}
-\frac{c_2}{q}\|u\|^q_{L^q}\\
 &\geq\frac{1}{p}\|u\|^p_{E^s}-\frac{1}{p^2}\|u\|^p_{E^s}
 -\frac{c_2}{q}\gamma^q_q\|u\|^q_{E^s}\\
 &\geq\frac{p-1}{p^2}\|u\|^p_{E^s}-\frac{c_2}{q}\gamma^q_q\|u\|^q_{E^s}.
   \end{align*}
Since $p<q$, the assertion follows.
\end{proof}

By (A1), there exists a constant $V_0>0$ such that
\[
  \widetilde{V}(x)=V(x)+V_0\geq1,\quad \forall x\in {\mathbb{R}^N}.
\]
Let
\begin{equation*}
  \widetilde{f}(x,u)=f(x,u)+V_0|u|^{p-2}u.
\end{equation*}
Then, it is easy to verify the following lemma.

\begin{lemma} \label{lem2.14}
Equation  \eqref{q}  is equivalent to the  problem
\begin{equation}\label{p41}
  (-\Delta)^s_pu+\widetilde{V}(x)|u|^{p-2}u=\widetilde{f}(x,u),
\quad forall x\in {\mathbb{R}^N}.
\end{equation}
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
Let $X=E^s$, $Y$ and $Z$ be defined by \eqref{p38} and \eqref{p39},
clearly, $\widetilde{f}$ satisfies (A3), (A5'), (A6) and (A7),
 Lemmas \ref{lem2.5}, \ref{lem2.6} and \ref{lem2.13} and
Corollary \ref{coro2.9} imply that $J$ satisfies
all conditions of Lemma \ref{lem2.4}. Thus,   \eqref{p41} possesses infinitely
many nontrivial solutions.
By Lemma \ref{lem2.14},  Equation \eqref{q} also possesses infinitely 
many nontrivial solutions.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.3}]
Let $X=E^s$, $Y$ and $Z$ be defined by \eqref{p38} and \eqref{p39},
clearly, $\widetilde{f}$ satisfies (A3), (A5'), (A6) and (A8),
Lemmas \ref{lem2.7} and \ref{lem2.13} and corollary \ref{coro2.9} imply that $J$ satisfies
all conditions of Lemma \ref{lem2.4}. Thus,  \eqref{p41} possesses infinitely
many nontrivial solutions.
By Lemma \ref{lem2.14},   \eqref{q} also possesses infinitely many nontrivial
solutions.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1.6}]
By \eqref{p}, \eqref{p4}, and (A7), we have
\begin{align*}
  J(u)&=\frac{1}{p}\|u\|^p_{E^s}-\int_{\mathbb{R}^N}F(x,u) dx\\
  &=\int_{\mathbb{R}^N}\left[\frac{1}{p}f(x,u)u-F(x,u)\right] dx
  \geq0,
  \end{align*}
and so $m=\inf_{u\in\mathcal{M}}J(u)\geq0$. We choose a sequence
$\{u_i\}\subset\mathcal{M}$ such that $J(u_i)\to  m$, as $i\to \infty$,
and $\|J'(u_i)\|(1+\|u_i\|)=0$. Hence, $\{u_i\}$ is a Cerami sequence,
there exists $u_0\in {E^s}$ such that $u_i\to  u_0$.
Since $J\in C^1(E^s,\mathbb{R})$, one has
\begin{equation*}
  J(u_0)=\lim_{i\to \infty}J(u_i)=m,\quad
J'(u_0)=\lim_{i\to \infty}J'(u_i).
\end{equation*}
Hence, we obtain that $u_0$ is also a critical point of $J$ and
$J(u_0)=\inf_{u\in\mathcal{M}}J(u)$. Furthermore, under assumptions
(A10) and (A3), we have
\begin{equation}\label{9}
  |f(x,u_i)|\leq \varepsilon|u_i|^{p-1}+C_\varepsilon|u_i|^{q-1},\quad
\forall \varepsilon>0
\end{equation}
for $2\leq p<q<p^*_s$.
By \eqref{9} and Lemma \ref{lem2.1}, we have
\begin{equation}\label{10}
\begin{aligned}
\|u_i\|^p_{E^s}
&=\int_{\mathbb{R}^N}f(x,u_i)u_i dx \\
&\leq \varepsilon\|u_i\|^p_{L^p}+C_\varepsilon\|u_i\|^q_{L^q}\\
&\leq\varepsilon\gamma_p\|u_i\|^p_{E^s}+C_\varepsilon\gamma_q\|u_i\|^q_{E^s}.
\end{aligned}
\end{equation}
For sufficiently small $\varepsilon>0$, \eqref{10} implies that there
exists a constant $\omega>0$ such that
$$
\|u_0\|_{E^s}=\lim_{i\to \infty}\|u_i\|_{E^s}\geq\omega>0.
$$
Thus, $u_0\neq0$.
\end{proof}

\subsection*{Acknowledgments}
This work was supported by the NNSF (Nos. 11601145, 11571370),
by the Fundamental Research Funds for the Central Universities of Central
South University (Grant No. 2017zzts312), by the Natural Science Foundation
of Hunan Provincial (No. 2017JJ3130), and by the Outstanding youth project
of Education Department of Hunan Province (17B143).

\begin{thebibliography}{99}

\bibitem{T26} A. Ambrosetti, P. H. Rabinowitz;
\emph{Dual variational methods in critical point theory and applications},
J. Funct. Anal. 14 (1973), 349-381.

\bibitem{VA} V. Ambrosio;
\emph{Multiple solutions for a fractional $p$-Laplacian equation with
sign-changing potential},  preprint arXiv:1603.05282, 2016-arxiv.org.

\bibitem{T11T} T. Bartolo, V. Benci, D. Fortunato;
\emph{Abstract critical point theorems and applications to some nonlinear
problems with strong resonance at infinity},
Nonlinear Anal. 7 (1983), 241-273.

\bibitem{T4} T. Bartsch, Z. Q. Wang, M. Willem;
\emph{The Dirichlet problem for superlinear elliptic equations},
in: M. Chipot, P. Quittner (Eds.), Handbook of Differential
Equations-Stationary Partial Differential Equations,
Vol. 2, Elsevier, 2005, pp. 1-5 (Chapter 1).

\bibitem{18CH}  X. Chang;
\emph{Ground state solutions of asymptotically linear fractional
Schr\"{o}dinger equations}, J. Math. Phys. 54 (2013), 061504.

\bibitem{17CH} X. Chang, Z. Wang;
\emph{Nodal multiple solutions of nonlinear problems involving
the fractional Laplacian}, J. Differ. Equ. 256 (2014), 2965-2992.

\bibitem{CHEN} B. T. Cheng, X. H. Tang;
\emph{New existence of solutions for the fractional $p$-Laplacian equations
with sign-changing potential and nonlinearity},
Mediterr. J. Math. 13 (2016), 3373-3387.

\bibitem{CM} M. Cheng;
\emph{Bound state for the fractional Schr\"{o}dinger equation with unbounded
potential}, J. Math. Phys. 53 (2012), 043507.

\bibitem{DI} S. Dipierro, G. Palatucci, E.  Valdinoci;
\emph{Existence and symmetry results for a Schr\"{o}dinger type problem
involving the fractional Laplacian}, Matematiche
68 (2013), 201-216.

\bibitem{11DONG} W. Dong, J. Xu, Z. Wei;
\emph{Infinitely many weak solutions for a fractional
Schr\"{o}dinger equation}, Boundary Value Prob. 159 (2014).

\bibitem{FP} P. Felmer, A. Quaas, J. Tan;
\emph{Positive solutions of nonlinear Schr\"{o}dinger
equation with the fractional Laplacian}. Proc. R. Soc. Edinburgh, Sect. A 142 (2012),
1237-1262.

\bibitem{X7}  T. Gou, H. Sun;
\emph{Solutions of nonlinear Schr\"{o}dinger equation with fractional
Laplacian without the Ambrosetti-Rabinowitz condition}, Appl. Math. Comput.
257 (2015), 409-416.

\bibitem{15HU} Y. Hua,  X. Yu;
\emph{On the ground state solution for a critical fractional Laplacian
equation}. Nonlinear Anal. 87 (2013), 116-125.

\bibitem{14TA} A. Iannizzotto,  S. Liu, F. Perera, M. Squassina;
\emph{Existence results for fractional $p$-Laplacian problems via Morse theory},
Adv. Calculus Var. 9 (2016), 101-125.

\bibitem{LJ} L. Jeanjean;
\emph{On the existence of bounded Palais-Smale sequences and application
to a Landesman-Lazer type problem set on $\mathbb{R}^N$},
Proc. Roy. Soc. Edinburgh 129 (1999), 787-809.

\bibitem{AS} N. Laskin;
\emph{Fractional quantum mechanics and L\'{e}vy path integrals},
Phys. Lett. A. 268 (2000), 298-305.

\bibitem{NL} N. Laskin;
\emph{ Fractional Schr\"{o}dinger equation}, Phys. Rev. E. 66 (2002), 056108.

\bibitem{LIN} X. Y. Lin, X. H. Tang;
\emph{Existence of infinitely many solutions for $p$-Laplacian equations
in $\mathbb{R}^N$}, Nonlinear Anal. 92 (2013), 72-81.

\bibitem{HL} H. Luo, X. Tang, S. Li;
\emph{Multiple solutions of nonlinear Schr\"{o}dinger equation
with the fractional $p$-Laplacian}, Taiwanese J. Math, preprint.

\bibitem{GB3} G. Molica Bisci;
\emph{Sequence of weak solutions for fractional equations},
Math. Res. Lett. 21 (2014), 241-253.

\bibitem{GM2} G. Molica Bisci, V. R\u{a}dulescu;
\emph{Ground state solutions of scalar field fractional Schr\"{o}dinger equations},
Calc. Var. Partial Differential Equations 54 (2015), 2985-3008.

\bibitem{GM1} G. Molica Bisci, D. Repov\v{s}, R. Servadei;
\emph{Nontrivial solutions of superlinear nonlocal problems},
Forum Math. 28 (2016), 1095-1110.

\bibitem{GB4} G. Molica Bisci, V. R\u{a}dulescu, R. Servadei;
\emph{Variational Methods for Nonlocal Fractional Problems},
Encyclopedia of Mathematics and its Applications, 162 Cambride
University Press-ISBN: 9781107111943. Foreword by J. Mawhin. (pp. 1-400).

\bibitem{DI7} E. D. Nezza, G. Palatucci, E. Valdinoci;
\emph{Hitchhikers guide to the fractional
Sobolev spaces}. Bull. Sci. Math. 136 (2012), 521-573.

\bibitem{T11P} P. H. Rabinowitz;
\emph{Minimax methods in critical point theory with applications to
differential equations}, in: CBMS Reg. Conf. Ser. in Math.,
vol. 65, Amer. Math. Soc., Providence, RI, 1986.

\bibitem{12SH} X. Shang, J. Zhang;
\emph{Concentrating solutions of nonlinear fractional
Schr\"{o}dinger equation with potentials}, J. Differ. Equ. 258 (2015), 1106-1128.

\bibitem{13Sh} X. Shang, J. Zhang;
\emph{On fractional Schr\"{o}dinger equation in $\mathbb{R}^N$ with critical
growth}, J. Math. Phys. 54 (2013), 121502.

\bibitem{SE} S. Secchi;
\emph{Ground state solutions for nonlinear fractional Schr\"{o}dinger equations
in $\mathbb{R}^N$}, J. Math. Phys. 54 (2013), 031501.

\bibitem{SEO} S. Secchi;
\emph{On fractional Schr\"{o}dinger equations in $\mathbb{R}^N$ without
 the Ambrosetti-Rabinowitz condition}, Topol.
Methods Nonlinear Anal. 47 (2016), 19-41.

\bibitem{21X8}  M. Souza;
\emph{On a class of nonhomogeneous fractional quasilinear equations
in $\mathbb{R}^N$ with exponential growth},
Nonlinear Differential Equations and Applications NoDEA, 4 (2015), 499-511.

\bibitem{T11CC} X. H. Tang;
\emph{Infinitely many solutions for semilinear Schr\"{o}dinger equations
with sign-changing potential and nonlinearity},
J. Math. Anal. Appl. 401(2013), 407-415.

\bibitem{TE}  K. Teng;
\emph{Multiple solutions for a class of fractional Schr\"{o}dinger equations in
$\mathbb{R}^N$}, Nonlinear Anal. Real World Appl. 21 (2015), 76-86.

\bibitem{TO} C. Torres;
\emph{On superlinear fractional $p$-Laplacian in $\mathbb{R}^N$}.
(2014), arXiv:1412.3392.

\bibitem{XJ} J. Xu, Z. We, W. Dong;
\emph{Weak solutions for a fractional $p$-Laplacian equation
with sign-changing potential}. Compl. Vari. Ellip. Equa. 61 (2015), 1-13.

\bibitem{10XU} J. Xu, Z. Wei, W. Dong;
\emph{Existence of weak solutions for a fractional
Schr\"{o}dinger equation}, Commun. Nonlinear Sci. Numer.
Simul. 22 (2015), 1215-1222.

\bibitem{ZB} B. L. Zhang, G. Molica Bisci, R. Servadei;
\emph{Superlinear nonlocal fractional problems with infinitely many solutions},
Nonlinearity 28 (2015), 2247-2264.

\bibitem{ZTZ} W. Zhang, X. Tang, J. Zhang;
\emph{Infinitely many radial and non-radial solutions for a fractional
 Schr\"{o}dinger equation}, Comput. Math. Appl. 71 (2016) 737-747.

\bibitem{ZZM} W. Zhang, J. Zhang, H. L. Mi;
\emph{On fractional Schr\"{o}dinger equation with periodic and
asymptotically periodic conditions}, Computers and Mathematics with Applications,
(2017), http://dx.doi.org/10.1016/j.camwa.2017.06.017

\end{thebibliography}

\end{document}