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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 80, pp. 1--19.\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-2017/80\hfil Ground state solutions]
{Ground state solutions for nonlinear fractional Schr\"odinger equations
involving critical growth}

\author[H. Jin, W. Liu \hfil EJDE-2017/80\hfilneg]
{Hua Jin, Wenbin Liu}

\address{Hua  Jin \newline
College of Science,
China University of Mining and Technology,
Xuzhou 221116, China}
\email{huajin@cumt.edu.cn}

\address{Wenbin Liu (corresponding author) \newline
College of Science,
China University of Mining and Technology,
Xuzhou 221116, China}
\email{liuwenbin-xz@163.com, phone (86-516) 83591530}

\dedicatory{Communicated by Marco Squassina}

\thanks{Submitted December 2, 2016. Published March 24, 2017.}
\subjclass[2010]{35A15, 35B33, 35Q55}
\keywords{Fractional Schr\"odinger equations; ground state solutions;
\hfill\break\indent  critical growth; Pohoz\u{a}ev identity}

\begin{abstract}
 This article concerns the ground state solutions of nonlinear fractional
 Schr\"odinger equations involving critical growth. We obtain the existence
 of ground state solutions when the potential is not a constant and not radial.
 We do not use the Ambrosetti-Rabinowitz condition, or the monotonicity
 condition on the nonlinearity.
\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}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

The fractional Laplacian $(-\Delta)^s$ is a classical linear integro-differential 
operator of order $s$. The main feature, and also its main difficulty, is that 
it is a non-local operator. Recently, a great deal of attention has been 
devoted to the fractional Laplacian and non-local operators of elliptic type, 
both for their interesting theoretical structure and concrete applications. 
The fractional Laplacian $(-\Delta)^s$ arises in the description of various 
phenomena in the applied science, such as the thin obstacle problem 
\cite{L.A.Caffarelli,Silvestre}, phase transition \cite{G.Alberti,Y.Sire},
 Markov processes \cite{Q.Y.Guan} and fractional quantum mechanics \cite{Laskin} 
and the references therein for more details.

The fractional Schr\"odinger equation formulated by
Laskin \cite{NLaskin1,NLaskin2,Laskin} has the  form
\begin{equation}
\label{fractionequation}
i \varphi_t-(-\Delta)^s\varphi-V(x)\varphi+f(\varphi)=0,\quad
 (x,t)\in \mathbb{R}^N\times\mathbb{R},
\end{equation}
where $s\in (0,1)$, $N>2s$, $\varphi$ is the  wavefunction and $V(x)$ is the
 potential energy. The fractional quantum mechanics has been discovered as a 
result of expanding the Feynman path integral, from
the Brownian-like to the L\'{e}vy-like quantum mechanical paths. 
Since we are concerned with the standing wave solutions of the form
\[
\varphi(x,t)=e^{-iwt}u(x), \quad w\in \mathbb{R},
\]
then  \eqref{fractionequation} can be converted into
\begin{equation} \label{p1}
(-\Delta)^su+V(x)u=f(u), \quad x\in \mathbb{R}^N.
\end{equation}
When $s=1$, equation \eqref{p1} gives back to the classical nonlinear 
Schr\"odinger equation
\begin{equation} \label{classicalequation}
-\Delta u+V(x)u=f(u), \quad x\in \mathbb{R}^N,
\end{equation}
which has been studied theoretically and numerically in the last decades.
We should emphasize that the potential $V(x)$ plays a crucial role concerning
the existence of nontrivial solutions and the existence of ground state solutions.
If the potential $V(x)$ is a constant, namely \eqref{classicalequation}
is autonomous, in the celebrated paper \cite{Berestycki},  Berestycki and
 Lions first proposed the Berestycki-Lions conditions which are almost
optimal for the existence of  ground state solutions in the subcritical case.
The authors investigated the constraint minimization problem and use the
Schwarz symmetrization in  $H_r^1(\mathbb{R}^N)$. For the critical nonlinearity $f$,
because of the lack of compactness of
$H^1(\mathbb{R}^N)\hookrightarrow L^{2^*}(\mathbb{R}^N)$,
 the existence of ground state solutions of problem \eqref{classicalequation}
becomes rather more complicated. In \cite{zjjzwm2}, the critical case was
considered by modifying the minimization methods with constrains.
 Since the radial symmetry plays a crucial role, the method is invalid for
the non-radial case.

In the  non-autonomous case, that is $V(x)\not\equiv V$, where $V$ is a constant, 
the main obstacle to get the existence of solutions or ground state solutions 
is the boundedness of the Palais-Smale (PS for short) sequence because of no 
some global conditions on $f$, such as the Ambrosetti-Rabinowitz (A-R for short) 
condition. Moreover, the lack of compactness due to the unboundedness of the 
domain prevents us from checking the (PS) condition.  
To avoid the difficulties mentioned above, in the seminal 
paper \cite{L.JeanjeanandKazunagaTanaka}, Jeanjean and Tanaka used an 
indirect approach developed in \cite{L.Jeanjean} to get a bounded 
(PS) (BPS for short) sequence for the energy functional $I$, then the existence 
of  positive solutions and moreover ground state solutions is obtained 
in the subcritical case when the nonlinearity $f$ and potential $V(x)$ 
satisfy the following assumptions:
\begin{itemize}

\item [(A1)] $f\in C(\mathbb{R}^+,\mathbb{R})$, $f(0)=0$ and $f'(0)$ defined as  
 $\lim_{t\to 0^+}f(t)/t$ exists,

\item [(A2)]  there is $p<\infty$ if $N=2$, $p<2^*-1$ if $N\geq 3$ such that 
$\lim_{t\to \infty}{f(t)/t^{p}}=0$,

\item [(A3)]  $\lim_{t\to \infty}{f(t)/t}=+\infty$,

\item [(A4)] $f'(0)<\inf\sigma(-\Delta+V(x))$, where $\sigma(-\Delta+V(x))$ denotes 
the spectrum of the self-adjoint operator $-\Delta+V(x)$,
\item [(A5)] $V\in C(\mathbb{R}^N,\mathbb{R})$, $V(x)\to V(\infty)\in \mathbb{R}$ as $|x|\to\infty$,
\item [(A6)] $V(x)\leq V(\infty)$,
\item [(A7)] there exists a function $\phi\in L^2(\mathbb{R}^N)\cap W^{1,\infty}(\mathbb{R}^N)$ such that
\[
|x||\nabla V(x)|\leq \phi^2(x), \forall x\in\mathbb{R}^N.
\]
\end{itemize}
Here, the decay condition (A7) is crucial to derive the boundedness of the 
(PS)-sequence. For the critical case, the problem is  different and more 
difficulty. In \cite{J.Zhang}, by use of the indirect approach developed in 
\cite{L.Jeanjean}, the authors completed the proof of existence of ground 
state solutions in the critical case with the same conditions on $V(x)$.
 As for the nonlinearity $f$, the following conditions are satisfied
\begin{itemize}

\item [(A8)] $f\in C(\mathbb{R}^+,\mathbb{R})$, $f(t)=o(t)$ as $t\to o^+$,

\item [(A9)]  $\lim_{t\to +\infty}f(t)/{t^{2^*-1}}=K>0$, where $2^*=\frac{2N}{N-2}$,

\item [(A10)]  there exist $D>0$ and $2<q<2^*$ such that 
 $f(t)\geq Kt^{2*-1}+Dt^{q-1},\forall t\geq 0$,

\item [(A11)] $f\in C^1(\mathbb{R}^+,\mathbb{R}),|f'(t)|\leq C(1+|t|^{\frac{4}{N-2}})$.
\end{itemize}

Now, we return our attention to the fractional and non-local problems.
 With the aid of the extended techniques developed
by Caffarelli and Silvestre \cite{Caffarelli}, some existence and 
nonexistence of Dirichlet problems involving the fractional Laplacian 
on bounded domains have been established, see \cite{Barriosa1,Cabre,J.Tan} 
and so on. For the general fractional Schr\"odinger equation
\[
(-\Delta)^su+V(x)u=f(x,u),x\in\mathbb{R}^N
\]
in the subcritical or critical case,  many results have been obtained on 
the existence of ground state solutions, positive solutions, 
the multiplicity of standing wave solutions, the symmetry of solutions 
and so forth, under the different conditions on $V(x)$ and $f$, for example 
the monotonicity condition, (A-R) condition, 
see \cite{B.Barriosa,bisci,Caffarelli,changxiaojun,chenzheng,tengkaiming,
weiyuanhong,zhangxia} and the references therein.

As is well known, the existence and concentration phenomena of solutions on 
the singularly perturbed fractional Schr\"odinger equation
\[
\varepsilon^{2s}(-\Delta)^su+V(x)u=f(u), x\in \mathbb{R}^N
\]
is also a hot topic. For this subject we refer, for example, 
to \cite{Alvesfrac,Davila,FMM,xiaominghe,seok,shangxzhangj} and the references 
therein.

Now, let us say more about the existence of  ground state solutions of a class 
of fractional scalar field equations
\[
(-\Delta)^su+V(x)u=f(u),x\in\mathbb{R}^N.
\]
When $f(u)-V(x)u=g(u)$, the authors \cite{changxiaojun2} obtained the existence 
of radial positive ground state solutions under the general Berestycki-Lions 
type assumptions in the case of subcritical growth. 
By using  the fractional Pohoz\v{a}ev identity and the monotonicity trick 
of Struwe-Jeanjean, they showed that the compactness still holds under 
their assumptions without the Strauss type radial lemma in $H_r^s(\mathbb{R}^N)$.
In \cite{zjjjm}, the existence of radial ground state solutions was obtained 
when $V(x)\equiv V$ involving the critical growth by means of the constraint 
variational argument, where $V>0$ is a constant. When $V(x)=V(|x|)$, 
Secchi \cite{simonesecchi}  proved the existence of radially symmetric solutions 
for equation \eqref{p1} in $H_r^s(\mathbb{R}^N)$ by the fractional Pohoz\v{a}ev
identity and the monotonicity trick in subcritical case. The conditions on $f$ 
and $V(x)$ are as follows
\begin{itemize}
\item [(A12)] $f\in C(\mathbb{R},\mathbb{R})$ is of class $C^{1,\gamma}$ for some $\gamma>\max\{0,1-2s\}$, 
and odd,

\item [(A13)]  $-\infty<\liminf_{t\to 0^+}{f(t)/t}\leq \limsup_{t\to 0^+}{f(t)/t}=-m<0$,

\item [(A14)]  $-\infty<\limsup_{t\to +\infty}{f(t)/{t^{2_s^*-1}}}\leq 0$,
 where $2_s^*=\frac{2N}{N-2s}$,

\item [(A15)] for some $\zeta>0$, there holds $F(\zeta)=\int_0^\zeta f(t)dt>0$,

\item [(A16)] $V\in C^1(\mathbb{R}^N,\mathbb{R}),V(x)\geq 0$ for every $x\in \mathbb{R}^N$ and
 this inequality is strict at some point,
\item [(A17)] $\|\max\{\langle\nabla V(x),x\rangle,0\}\|_{L^{N/2s}(\mathbb{R}^N)}<2sS_s$,
\item [(A18)] $\lim_{|x|\to +\infty}V(x)=0$,
\item [(A19)] $V(x)$ is radially symmetric,
\end{itemize}
where $S_s$ is the best Sobolev constant for the critical embedding, that is
\[
S_s=\inf_{u\in H^s(\mathbb{R}^N),{u\neq 0}}
\frac{\|(-\Delta)^{s/2}u\|^2_{L^2}}{\|u\|^2_{L^{2_s^*}}},
\]
here $H^s(\mathbb{R}^N)$ is the fractional Sobolev space with respect to the norm
\[
\|u\|^2=\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2+|u|^2.
\]
Where (A13)-(A15) are called Berestycki-Lions type conditions, and 
(A17) is used to get the boundedness of the (PS)-sequence by use of the 
 monotonicity trick. The condition $f\in C^1$ ensures that the fractional 
Pohoz\v{a}ev identity can be used.

Now, the problem is how about the existence of ground state solutions 
when $V(x)$ is non-radial in the critical case. As we all know, for the 
critical case, the loss of the compactness for the embedding 
$H^s(\mathbb{R}^N)\hookrightarrow L^{2_s^*}(\mathbb{R}^N)$ is the main 
difficulty. What's more, (PS) condition, in general, fails. 
Since $V(x)$ is non-radial, the method introduced in \cite{zjjjm} 
can not be used here since the fractional space they used is $H_r^s(\mathbb{R}^N)$.
 With critical growth, the authors \cite{xiaominghe} proved the existence 
of solutions for equation \eqref{p1} under the monotonicity condition 
on $f(t)/t$  and  $0<\mu F(t)=\mu \int_0^tf(t)dt\leq tf(t),\mu\in(2,2_s^*)$ 
for all $t>0$.

Motivated by the seminal papers above, we use the indirect approach developed 
in \cite{L.Jeanjean} to investigate the existence of ground state solutions 
for nonlinear  fractional Schr\"odinger equation \eqref{p1} involving
the critical nonlinearity, where the potential $V(x)$ depends on $x$ non-radially.
More precisely, on the nonlinearity $f$, we assume
\begin{itemize}

\item [(A20)] $f\in C^1(\mathbb{R}^+,\mathbb{R})$ and $\lim_{t\to 0}f(t)/t=0$,

\item [(A21)]  $\lim_{t\to \infty}{f(t)/t^{2_s^*-1}}=1$,

\item [(A22)]  There exist $D>0$ and $p<2_s^*$ such that 
$f(t)\geq t^{2_s^*-1}+Dt^{p-1},t\geq 0$.

\end{itemize}
We assume $f(t)\equiv 0$ for $t\leq 0$  throughout the paper since we are 
concerned with the positive solutions.

On potential $V\in C^1(\mathbb{R}^N,\mathbb{R})$, we assume
\begin{itemize}
\item [(A23)] There exists $V_0>0$ such that $\inf_{x\in \mathbb{R} ^N}V(x)\geq V_0$,
\item [(A24)] $V(x)\leq V(\infty):=\lim_{|x|\to \infty}V(x)<\infty$ for all $x\in \mathbb{R}^N$ and $V(x)\not \equiv  V(\infty)$,
\item [(A25)] $\|\max\{\langle\nabla V(x),x\rangle,0\}\|_{L^{N/2s}(\mathbb{R}^N)}<2sS_s$.
\end{itemize}

In contrast to the conditions in \cite{xiaominghe}, our conditions are more weaker.
The main result is  the following.

\begin{theorem} \label{thm1.1}
Assume $N>2s,s\in(0,1)$, if $\max\{2,2_s^*-2\}<p<2_s^*$, {\rm (A20)--(A25)} 
hold, then  problem \eqref{p1} has a ground state solution.
\end{theorem}

The proof of Theorem \ref{thm1.1} is inspired by the ideas 
in \cite{L.JeanjeanandKazunagaTanaka} and \cite{J.Zhang}.

Firstly, we show the existence of positive solutions of $\eqref{p1}$.
 For this purpose, we look for a special BPS sequence for the energy functional
 $I$ associated with $\eqref{p1}$ by use of the Struwe's monotonicity trick. 
Precisely, with the help of the auxiliary energy functional $I_\lambda$ satisfying
\[
I(u_{\lambda_j})=I_{\lambda_j}(u_{\lambda_j})+(\lambda_j-1)\int_{\mathbb{R}^N} F(u_{\lambda_j}),\lambda_j\to 1,\quad
 j\to \infty,
\]
we prove the existence of positive critical points denoted by $u_{\lambda_j}$ of 
$I_{\lambda_j}$. Thanks to the decomposition of BPS sequence, the properties 
of $\{u_{\lambda_j}\}$ and the energy estimation of $I_{\lambda_j}(u_{\lambda_j})$ are obtained. 
Consequently, we show that $\{u_{\lambda_j}\}$ is a BPS sequence for $I$ at some 
level value.

Secondly, for the proof of the existence of ground state solutions, 
we construct a minimizing sequence $\{u_n\}$ which is composed of the 
critical points of $I$. We show that $\{u_n\}$ is a BPS sequence for $I$ at $m$, 
here $m$ denotes the least energy. Then, making use of the decomposition of 
BPS sequence and the relationship of $I$ and $I^\infty$, we prove that $m$ is 
attained at some $\tilde u\neq 0$.

\begin{remark} \label{rmk1.2} \rm 
In the proof of our main results, the estimations of the Mountain Pass 
(MP for short) values, Pohoz\v{a}ev identity and the decomposition of
 BPS all play crucial roles.
\end{remark}

This article is organized as follows. 
In section 2, we introduce a variational setting of our problem and 
present some preliminary results. 
In section 3, we are concerned with the decomposition of BPS and the
 existence of nontrivial critical points for the auxiliary energy functional. 
Section 4 is devoted to the completion of the proof of Theorem \ref{thm1.1}.

In the following, the letters $C,\delta, \delta_0$ are indiscriminately used 
to denote various positive constants whose exact values are irrelevant.

\section{Preliminaries and functional setting}

To establish the variational setting for \eqref{p1}, we give some useful 
facts of the fractional Sobolev space \cite{Nezza} and some preliminary lemmas.

The fractional Laplacian operator $(-\Delta)^s$ with $s\in(0,1)$ of a function 
$u:\mathbb{R}^N\rightarrow \mathbb{R}$ is defined by
\[
\mathcal{F}((-\Delta)^s u)(\xi)=|\xi|^{2s}\mathcal{F}(u)(\xi),\quad \xi\in\mathbb{R}^N,
\]
where $\mathcal{F}$ is the Fourier transform. For $s\in (0,1)$, the fractional 
order Sobolev space $H^s(\mathbb{R}^N)$ is defined by
\[
H^s(\mathbb{R}^N)=\{u\in L^2(\mathbb{R}^N):
\int_{\mathbb{R}^N}|\xi|^{2s}|\hat{u}|^2d\xi<\infty\},
\]
endowed with the norm $\|u\|_{H^s(\mathbb{R}^N)}
=(\int_{\mathbb{R}^N}(|\xi|^{2s}|\hat{u}|^2+|\hat{u}|^2)d\xi)^{1/2}$, where
$\hat{u}\doteq \mathcal{F}(u)$.
By Plancherel's theorem, we have 
$\|u\|_{L^2(\mathbb{R}^N)}=\|\hat{u}\|_{L^2(\mathbb{R}^N)}$ and
\[
\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u(x)|^2dx
=\int_{\mathbb{R}^N}(|\xi|^s|\hat{u}|)^2d\xi.
\]
It follows that $\|u\|_{H^s(\mathbb{R}^N)}
=\big(\int_{\mathbb{R}^N}(|(-\Delta)^{s/2}u(x)|^2+|u|^2)dx\big)^{1/2}$,
$u\in H^s(\mathbb{R}^N)$.
If $u$ is smooth enough, $(-\Delta)^s u$ can be computed by the following singular 
integral
\[
(-\Delta)^s u(x)=c_{N,s}\text{P.V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy.
\]
Here $c_{N,s}$ is the normalization constant and P.V. is the principal value. 
So, one can get an alternative definition of the fractional Sobolev space 
$H^s(\mathbb{R}^N)$ as follows,
\[
H^s(\mathbb{R}^N)=\{u\in L^2(\mathbb{R}^N):\frac{|u(x)-u(y)|}{|x-y|^{\frac{N+2s}{2}}}
\in L^2(\mathbb{R}^N\times\mathbb{R}^N)\}��
\]
with the norm
\[
\|u\|_{H^s(\mathbb{R}^N)}
=\Big(\int_{\mathbb{R}^N}|u|^2+\int_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\Big)^{1/2}.
\]
The space
$D^s(\mathbb{R}^N)$ denotes the completion of $C_0^\infty(\mathbb{R}^N)$ with respect
to the Gagliardo norm
\[
\|u\|_{D^s(\mathbb{R}^N)}
=\Big(\int_{\mathbb{R}^N}|\xi|^{2s}|\hat{u}|^2d\xi\Big)^{1/2}
=\Big(\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2\Big)^{1/2}.
\]
 Since we investigate the existence of solutions of  problem  \eqref{p1}, 
we need the fractional Sobolev space  $H_{V}^s(\mathbb{R}^N)$ which is a 
Hilbert subspace of $H^s(\mathbb{R}^N)$ with the norm
\begin{equation} \label{norm}
\|u\|_{H_{V}^s(\mathbb{R}^N)}
:=\Big(\int_{\mathbb{R}^N}\left(|(-\Delta)^{s/2}u|^2+V(x)|u|^2\right)dx
\Big)^{1/2}<\infty.
\end{equation}
It is easy to check that $H_{V}^s(\mathbb{R}^N)\equiv H^s(\mathbb{R}^N)$ if
(A23) and (A24) hold. In our paper, we shall work on $H^s(\mathbb{R}^N)$ with
norm \eqref{norm}
and we denote $\|u\|_{H^s(\mathbb{R}^N)}$ by $\|u\|$ for simplicity.

Associated with problem \eqref{p1}, is the energy functional 
$I:H^s(\mathbb{R}^N)\to \mathbb{R}$ defined by
\[
I(u)=\frac{1}2\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2+V(x)|u|^2-\int_{\mathbb{R}^N}F(u),u\in H^s(\mathbb{R}^N),
\]
where $F(u)=\int_0^uf(t)dt$. Conditions (A20)--(A22)
imply that $I\in C^1(H^s(\mathbb{R}^N),\mathbb{R})$.

\begin{definition} \label{def2.1} \rm 
$u$ is said to be a solution of $\eqref{p1}$ if $u$ is a critical point of
 the energy functional $I$ and satisfies
\[
\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}}u(-\Delta)^{\frac{s}{2}}\varphi+\int_{\mathbb{R}^N}V(x)u\varphi
=\int_{\mathbb{R}^N}f(u)\varphi,u\in H^s(\mathbb{R}^N),\quad \forall \varphi\in C_0^\infty(\mathbb{R}^N).
\]
$u$ is said to be a ground state solution of \eqref{p1} if $u$ is a solution 
with the least energy among all nontrivial solutions of \eqref{p1}.
\end{definition}

In this article, we use the embedding lemma and  Lions lemma  as follows.

\begin{lemma} \label{lemma1.1}(\cite{lionsembedding})
For any $s\in(0,1)$, $H^s(\mathbb{R}^N)$ is continuously embedded into $L^r(\mathbb{R}^N)$ for $r\in [2,2_s^*]$ and compactly embedded into $L_{loc}^r(\mathbb{R}^N)$ for $r\in [2,2_s^*)$.
\end{lemma}

\begin{lemma} \label{lionslemma}(\cite{lionslemma})
Suppose that $\{u_n\}$ is bounded in $H^s(\mathbb{R}^N)$ and
\[
\lim_{n\to \infty}\sup_{z\in \mathbb{R}^N}\int_{B_1(z)}|u_n|^2\to 0.
\]
Then $\|u_n\|_{L^r}\to 0$ for $r\in (2,2_s^*)$ when $N\geq3$ and for $r\in (2,+\infty)$ when $N=1,2$. Here $B_1(z)=\{y\in\mathbb{R}^N,|y-z|\leq1\}$.
\end{lemma}




\section{Solutions for auxiliary problems}

In this section, we consider the family of functionals 
$I_\lambda(u):H^s(\mathbb{R}^N)\to \mathbb{R}$ defined by
\[
I_\lambda(u)=\frac{1}2\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2+V(x)|u|^2
-\lambda\int_{\mathbb{R}^N}F(u).
\]
The corresponding auxiliary problems are
\begin{equation} \label{auxiliaryproblem}
(-\Delta)^su+V(x)u=\lambda f(u).
\end{equation}
The main aim of this section is to prove that for almost every $\lambda\in[1/2,1]$, 
$I_\lambda$ has a nontrivial critical point $u_\lambda$ such that $I_\lambda(u_\lambda)\leq c_\lambda$, 
where
\begin{gather*}
c_\lambda=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I_\lambda(\gamma(t)), \\
\Gamma=\{\gamma\in C([0,1],H^s(\mathbb{R}^N)),\gamma(0)=0 \ \ \text{and}\ \ I_\lambda(\gamma(1))<0\}.
\end{gather*}

Before we prove the existence of solutions for the auxiliary problems 
\eqref{auxiliaryproblem}, we give some propositions and lemmas.

\begin{proposition}  %\label{pohozaev identity}
Let $u(x)$ be a critical point of $I_\lambda$ with $\lambda\in[1/2,1]$, 
then $u(x)$ satisfies
\begin{equation} \label{pohozaev identity}
\begin{aligned}
&\frac{N-2s}2\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2
+\frac{N}2\int_{\mathbb{R}^N}V(x)|u|^2 \\
&+\frac{1}2\int_{\mathbb{R}^N}
\langle \nabla V(x),x\rangle |u|^2-N\lambda\int_{\mathbb{R}^N}F(u)=0.
\end{aligned}
\end{equation}
\end{proposition}
As we all know, \eqref{pohozaev identity} is named Pohoz\v{a}ev identity. 
The proof is similar as that in \cite{changxiaojun2} and we omit it here.

\begin{lemma} \label{splittinglemma} 
Assume {\rm (A20)} and {\rm (A21)} hold. 
Let $\{u_n\}\subset H^s(\mathbb{R}^N)$ be such that $u_n\to u$ 
weakly in $H^s(\mathbb{R}^N)$. Then up to a subsequence,
\[
\int_{\mathbb{R}^N}(f(u_n)-f(u)-f(u_n-u))\phi=o_n(1)\|\phi\|.
\]
where $o_n(1)\to 0$ uniformly for $\phi\in C_0^\infty(\mathbb{R}^N)$ as $n\to \infty$.
\end{lemma}

The proof of the above lemma is  similar to that in \cite{zjjzwm}. 
So we omit it. 
Similar the proof of Brezis-Lieb Lemma in \cite{brezis-lieb}, 
we can give the following lemma.

\begin{lemma} \label{Brezis-Lieb lemma} 
For $s\in (0,1)$, assume {\rm (A20)} and {\rm(A21)}. 
Let $\{u_n\}\subset H^s(\mathbb{R}^N)$ such that $u_n\to u$ weakly 
in $H^s(\mathbb{R}^N)$
and a.e. in $\mathbb{R}^N$ as $n\to \infty$, then
\[
\int_{\mathbb{R}^N}F(u_n)=\int_{\mathbb{R}^N}F(u_n-u)+\int_{\mathbb{R}^N}F(u)+o_n(1),
\]
where $o_n(1)\to 0$ as $n\to \infty$.
\end{lemma} 

To obtain the existence of critical points for $I_\lambda$, the following abstract 
result is needed from \cite{L.Jeanjean}, which shows that for almost every  
$\lambda\in[1/2,1]$, $I_\lambda$ possesses a BPS  sequence at the level $c_\lambda$.

\begin{theorem} \label{thm2.1} 
Let $X$ be a Banach space equipped with a norm $\|\cdot\|_X$ and let 
$J\subset\mathbb{R}^+$ be an interval. For a family $(I_\lambda)_{\lambda\in J}$ of 
$C^1$-functionals on $X$ of the form
\[
I_\lambda(u)=A(u)-\lambda B(u),\forall \lambda\in J,
\]
where $B(u)\geq 0,\forall u\in X$ and such that either $A(u)\to +\infty$ or 
$B(u)\to +\infty$ as $\|u\|_X\to \infty$.
If there are two points $v_1,v_2$ in $X$ such that
\[
c_\lambda=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I_\lambda(\gamma(t))>\max\{I_\lambda(v_1),I_\lambda(v_2)\},\quad
\forall \lambda\in J,
\]
where
\[
\Gamma=\{\gamma\in C([0,1],X),\gamma(0)=v_1,\gamma(1)=v_2\}.
\]
Then, for almost every $\lambda\in J$, there is a sequence $\{v_n\}\subset X$ such that
\begin{itemize}
\item[(i)] $\{v_n\}$ is bounded,
\item[(ii)] $I_\lambda(v_n)\to c_\lambda$,
\item[(iii)] $I'_\lambda(v_n)\to 0$ in  the dual $X^{-1}$  of $X$.
\end{itemize}
\end{theorem}

In the following, we use Theorem \ref{thm2.1} to seek nontrival critical 
points of $I_\lambda$ for almost every $\lambda\in J$. In what follows, 
let $X=H^s(\mathbb{R}^N)$ and
\[
A(u)=\frac{1}2\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2+V(x)|u|^2,\quad
 B(u)=\int_{\mathbb{R}^N}F(u).
\]
Obviously, $A(u)\to +\infty$ as $\|u\|\to \infty$ and $B(u)\geq 0$ for any
 $u\in H^s(\mathbb{R}^N)$ by (A22). Now, we give the following lemma 
to ensure that
$I_\lambda$ has the MP geometry. Consequently, we obtain a BPS
for $I_\lambda$ by Theorem \ref{thm2.1}.

\begin{lemma} \label{Lemma2.2}
Assume {\rm (A20)--(A24)} hold. Then
\begin{itemize}
\item[(i)] there exists a $v\in H^s(\mathbb{R}^N)\setminus\{0\}$ with
$I_\lambda(v)\leq 0$ for all $\lambda\in [1/2,1]$;

\item[(ii)] $c_\lambda=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I_\lambda(\gamma(t))
>\max\{I_\lambda(0),I_\lambda(v)\}>0$ for all $\lambda\in [1/2,1]$,
 where
\[
\Gamma=\{\gamma\in C([0,1],H^s(\mathbb{R}^N)),\gamma(0)=0,\gamma(1)=v\};
\]

\item[(iii)] there exists a BPS sequence $\{u_n\}$ at the MP level 
$c_\lambda$ for $I_\lambda$, where  $u_n\geq 0$.
\end{itemize}
\end{lemma}

\begin{proof} Since (A20) and (A21) hold, for any $\varepsilon>0$, 
there exists $C(\varepsilon)>0$ such that
\[
\int_{\mathbb{R}^N}F(u)\leq \varepsilon\int_{\mathbb{R}^N}|u|^2+C(\varepsilon)\int_{\mathbb{R}^N}|u|^{2_s^*},\forall
u\in H^s(\mathbb{R}^N).
\]
Thus
\begin{align*}
I_\lambda(u)
&=\frac{1}2\|u\|^2-\lambda\int_{\mathbb{R}^N}F(u)\\
&\geq \frac{1}2\|u\|^2-\varepsilon\|u\|_{L^2}^2-C(\varepsilon)\|u\|_{L^{2_s^*}}^{2_s^*}
\end{align*}
From Lemma \ref{lemma1.1}, there exist constants $\rho>0$  and $\delta>0$ 
independent of $\lambda$ such that for $\|u\|=\rho$, $I_\lambda(u)\geq \delta$.
On the other hand, (A22) implies 
\[
I_\lambda(u)\leq\frac{1}2\|u\|^2-\frac{1}2\|u\|_{L^{2_s^*}}^{2_s^*}
-\frac{D}{2p}\|u\|_{L^p}^p.
\]
Set $v_0\in H^s(\mathbb{R}^N)$ such that $v_0\geq 0,v_0\neq 0$.
Since $I_\lambda(tv_0)\to -\infty$ as $t\to+\infty$, then there exists $t_0$ such that 
$I_\lambda(t_0v_0)<0$ as $\|t_0v_0\|>\rho$. Set $v=t_0v_0$, then $(i)$ and $(ii)$ hold.
So, the conditions of Theorem \ref{thm2.1} are satisfied. 
Therefore, for almost every $\lambda\in [1/2,1]$, there exists a BPS sequence
$\{u_n\}$ for $I_\lambda$ at the MP value $c_\lambda$.
 Now, we show $u_n\geq 0$. Let $u_n=u_n^++u_n^-$. Using $u_n^-$ as a test function, 
since $f(t)\equiv 0$ for all $t\leq 0$, we have
\begin{align*}
(I'_\lambda(u_n),u_n^-)
&=\int_{\mathbb{R}^N}(-\Delta)^{s/2}u_n(-\Delta)^{s/2}u_n^-
 +\int_{\mathbb{R}^N}V(x)(u_nu_n^-)-\lambda\int_{\mathbb{R}^N}f(u_n)u_n^-\\
&=\int_{\mathbb{R}^N}(-\Delta)^{s/2}u_n(-\Delta)^{s/2}u_n^-
 +\int_{\mathbb{R}^N}V(x)|u_n^-|^2.
\end{align*}
Since for every $x,y\in\mathbb{R}^N$, we  have
$\left(u_n^+(x)-u_n^+(y)\right)\left(u_n^-(x)-u_n^-(y)\right)\geq 0$, it follows
that
\begin{align*}
&\left(u_n(x)-u_n(y)\right)\left(u_n^-(x)-u_n^-(y)\right) \\
&=\left(u_n^+(x)-u_n^+(y)\right)\left(u_n^-(x)-u_n^-(y)\right)+\left(u_n^-(x)-u_n^-(y)\right)^2\\
&\geq \left(u_n^-(x)-u_n^-(y)\right)^2.
\end{align*}
Thus
\begin{align*}
\int_{\mathbb{R}^N}(-\Delta)^{s/2}u_n(-\Delta)^{s/2}u_n^-
&=\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\left(u_n(x)-u_n(y)\right)
 \left(u_n^-(x)-u_n^-(y)\right)}{|x-y|^{N+2s}}\\
&\geq \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\left(u_n^-(x)-u_n^-(y)\right)^2}{|x-y|^{N+2s}}\\
&=\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_n^-|^2.
\end{align*}
Therefore, from $(I'_\lambda(u_n),u_n^-)\to 0$, we have $\|u_n^-\|\to 0$. 
The proof is complete.
\end{proof}

From the argument above, we obtain a BPS for $I_\lambda$ at the level $c_\lambda$. 
To get the convergence of the BPS sequence, we give some lemmas and propositions.

\begin{lemma} \label{Lemma2.3}
Assume {\rm (A20)--(A24)} hold.
If $\max\{2,2_s^*-2\}<p<2_s^*$, then
\[
c_\lambda<\frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}.
\]
\end{lemma}

\begin{proof}  
Let $\varphi\in C_0^\infty(\mathbb{R}^N)$ is a cut-off function with support $B_2$ such that
$\varphi\equiv1$ on $B_1$ and $0\leq \varphi\leq1$ on $B_2$, where $B_r$ denotes the 
ball in $\mathbb{R}^N$ of center at origin and radius $r$. For $\varepsilon>0$, we define
$\psi_\varepsilon(x)=\varphi(x)U_\varepsilon(x)$, where
\[
U_\varepsilon(x)=\kappa\varepsilon^{-\frac{N-2s}2}\Big(\mu^2+\big|\frac{x}{\varepsilon S_s^{\frac{1}{2s}}}
\big|^2\Big)^{-\frac{N-2s}2}.
\]
By \cite{A.Cotsiolis}, $S_s$ can be achieved by $U_\varepsilon(x)$. 
Let $v_\varepsilon=\frac{\psi_\varepsilon}{\|\psi_\varepsilon\|_{L^{2_s^*}}}$, then 
$\|(-\Delta)^{s/2}v_\varepsilon\|_{L^2}^2\leq S_s+O(\varepsilon^{N-2s})$.
From \cite{xiaominghe}, we have the estimates
\[
\|v_\varepsilon\|_{L^2}^2=\begin{cases}
O(\varepsilon^{2s}),& N>4s,\\
O(\varepsilon^{2s}\ln\frac{1}\varepsilon), & N=4s,\\
O(\varepsilon^{N-2s}),& N<4s,
\end{cases}
\]
and
\[
\|v_\varepsilon\|_{L^p}^p=\begin{cases}
O(\varepsilon^{\frac{2N-(N-2s)p}2}), & p>\frac{N}{N-2s},\\
O(\varepsilon^{\frac{(N-2s)p}2}),    &  p<\frac{N}{N-2s}.
\end{cases}
\]
By (A22), for any  $t>0$,
\begin{align*}
I_\lambda(tv_\varepsilon)
&=\frac{t^2}2\int_{\mathbb{R}^N}|(-\Delta)^{s/2}v_\varepsilon|^2+V(x)|v_\varepsilon|^2
 -\lambda\int_{\mathbb{R}^N} F(tv_\varepsilon)\\
&=\frac{t^2}2\|v_\varepsilon\|^2-\lambda\int_{\mathbb{R}^N} F(tv_\varepsilon)\\
&\leq \frac{t^2}2\|v_\varepsilon\|^2-\frac{\lambda}{2_s^*}t^{2_s^*}
 -\frac{Dt^p}{2p}\|v_\varepsilon\|_{L^p}^p.
\end{align*}
Obviously, $I_\lambda(tv_\varepsilon)\to -\infty$ as $t\to +\infty$ and $I_\lambda(tv_\varepsilon)>0$ for $t>0$
small.
 Let $g(t)=\frac{t^2}2\|v_\varepsilon\|^2-\frac{\lambda}{2_s^*}t^{2_s^*}$. 
Then $t_\varepsilon=\big(\frac{\|v_\varepsilon\|^2}\lambda\big)^{\frac{1}{2_s^*-2}}$ is the maximum 
point of $g(t)$.

For $\varepsilon<1$, by the definition of $v_\varepsilon$, there exists $t_1>0$ small enough such that
\[
\max_{t\in (0,t_1)}I_\lambda(tv_\varepsilon)\leq \frac{t^2}2\|v_\varepsilon\|^2
<\frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}.
\]
Since $I_\lambda(tv_\varepsilon)\to -\infty$ as $t\to +\infty$, it is easy to obtain that there 
exists $t_2>0$ such that
\[
\max_{t\in (t_2,+\infty)}I_\lambda(tv_\varepsilon)
<\frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}.
\]
If $t\in [t_1,t_2]$,
\[
\max_{t\in [t_1,t_2]}I_\lambda(tv_\varepsilon)
\leq \max_{t\in [t_1,t_2]}\{g(t)-\frac{Dt_1^p}{2p}\|v_\varepsilon\|_{L^p}^p\}  
\leq g(t_\varepsilon)-\frac{Dt_1^p}{2p}\|v_\varepsilon\|_{L^p}^p
\]
For $g(t_\varepsilon)$, we have
\begin{align*}
g(t_\varepsilon)
&=\frac{s}{N\lambda^{\frac{N-2s}{2s}}}(\|v_\varepsilon\|^2)^{\frac{N}{2s}}\\
&=\frac{s}{N\lambda^{\frac{N-2s}{2s}}}
 \Big(\|(-\Delta)^{s/2}v_\varepsilon\|_{L^2}^2+\int_{\mathbb{R}^N}V(x)|v_\varepsilon|^2\Big)^{\frac{N}{2s}}\\
&\leq \frac{s}{N\lambda^{\frac{N-2s}{2s}}}\Big(S_s+O(\varepsilon^{N-2s})+C\|v_\varepsilon\|_{L^2}^2
\Big)^{\frac{N}{2s}}.
\end{align*}
By $(a+b)^q\leq a^q+q(a+b)^{q-1}b$, where $a>0,b>0,q>1$, we have
\begin{align*}
g(t_\varepsilon)
&\leq \frac{s}{N\lambda^{\frac{N-2s}{2s}}}
 \Big(S_s^{\frac{N}{2s}}+{\frac{N}{2s}}\Big(S_s+O(\varepsilon^{N-2s})\\
&\quad +C\|v_\varepsilon\|_{L^2}^2\Big)^{\frac{N-2s}{2s}}
 \left(O(\varepsilon^{N-2s})+C\|v_\varepsilon\|_{L^2}^2\right)\Big)\\
&\leq \frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}+O(\varepsilon^{N-2s})
 +C\|v_\varepsilon\|_{L^2}^2.
\end{align*}
 Thus
\[
\max_{t\in [t_1,t_2]}I_\lambda(tv_\varepsilon)\leq \frac{s}{N\lambda^{\frac{N-2s}{2s}}}
 S_s^{\frac{N}{2s}}+O(\varepsilon^{N-2s})+C\|v_\varepsilon\|_{L^2}^2-\frac{Dt_1^p}{2p}\|v_\varepsilon\|_{L^p}^p.
\]
Next, we estimate $\max_{t\in [t_1,t_2]}I_\lambda(tv_\varepsilon)$ in three cases.
\smallskip

\noindent\textbf{Case 1:} If $N>4s$, then $\frac{N}{N-2s}<2$,  with 
$p>\max\{2,2_s^*-2\}$, we have $p>\frac{N}{N-2s}$. So
\[
\max_{t\in [t_1,t_2]}I_\lambda(tv_\varepsilon)\leq \frac{s}{N\lambda^{\frac{N-2s}{2s}}}
S_s^{\frac{N}{2s}}+O(\varepsilon^{N-2s})+O(\varepsilon^{2s})-O(\varepsilon^{\frac{2N-(N-2s)p}2}).
\]
From $p>2,N>4s$, then $\frac{2N-(N-2s)p}2<2s<N-2s$. Thus, for $\varepsilon>0$ small enough, 
we obtain
\[
\max_{t\in [t_1,t_2]}I_\lambda(tv_\varepsilon)< \frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}.
\]

\noindent\textbf{Case 2:} If $N=4s$, then $2<p<4$. For $\varepsilon>0$ small enough, 
we obtain
\begin{align*}
\max_{t\in [t_1,t_2]}I_\lambda(tv_\varepsilon)
&\leq \frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}
 +O(\varepsilon^{N-2s})+O(\varepsilon^{2s}\ln\frac{1}\varepsilon)-O(\varepsilon^{4s-sp})\\
&\leq\frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}
 +O\Big(\varepsilon^{2s}(1+\ln\frac{1}\varepsilon)\Big)-O(\varepsilon^{4s-sp})\\
&<\frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}},
\end{align*}
since
\[
\lim_{\varepsilon\to 0^+}\frac{\varepsilon^{4s-sp}}{\varepsilon^{2s}(1+\ln\frac{1}\varepsilon)}\to +\infty.
\]

\noindent\textbf{Case 3:} If $2s<N<4s$, then $\frac{N}{N-2s}>2$, 
 with $p>\max\{2,2_s^*-2\}$, we have $p>\frac{N}{N-2s}$. So
\[
\max_{t\in [t_1,t_2]}I_\lambda(tv_\varepsilon)\leq \frac{s}{N\lambda^{\frac{N-2s}{2s}}}
S_s^{\frac{N}{2s}}+O(\varepsilon^{N-2s})-O(\varepsilon^{\frac{2N-(N-2s)p}2}).
\]
From $p>\frac{4s}{N-2s}$, then $\frac{2N-(N-2s)p}2<N-2s$. For $\varepsilon>0$ small
 enough, we obtain
\[
\max_{t\in [t_1,t_2]}I_\lambda(tv_\varepsilon)< \frac{s}{N\lambda^{\frac{N-2s}{2s}}}
S_s^{\frac{N}{2s}}.
\]
The proof is complete.
\end{proof}

In \eqref{p1}, if $V(x)\equiv V(\infty)$, for $\lambda \in [1/2,1]$,
the family of functionals $I_\lambda^\infty:H^s(\mathbb{R}^N)\mapsto \mathbb{R}$, defined as
\[
I_\lambda^\infty(u)=\frac{1}2\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2+V(\infty)|u|^2
-\lambda\int_{\mathbb{R}^N}F(u),
\]
plays an important role in our paper.
Similar  as that in \cite{JeanTa,zjjzwm2}, we can derive the following result.

\begin{lemma} \label{Lemma2.5}
 For $\lambda \in [1/2,1]$, if $w_\lambda\in H^s(\mathbb{R}^N)$ is a nontrivial critical
point of $I_\lambda^\infty$, then there exists $\gamma_\lambda\in C([0,1],H^s(\mathbb{R}^N))$ such that
$\gamma_\lambda(0)=0,I_\lambda^\infty(\gamma_\lambda(1))<0,w_\lambda\in \gamma_\lambda[0,1]$ and
 $\max_{t\in[0,1]}I_\lambda^\infty(\gamma_\lambda(t))=I_\lambda^\infty(w_\lambda)$.
\end{lemma}

\begin{lemma}[\cite{zjjjm}] \label{Lemma2.6}
If $f$ satisfies {\rm (A20)--(A22)} and $\max\{2,2_s^*-2\}<p<2_s^*$, then 
for almost every $\lambda \in [1/2,1]$, $I_\lambda^\infty$ has a positive ground
state solution.
\end{lemma}

\begin{lemma} \label{energyest}
If $V(x)\equiv V(\infty)>0$ and {\rm (A20)} and {\rm (A21)} hold, then there exists a 
constant $\delta> 0$ independent of $\lambda$ such that any nontrivial critical 
point $u$ of $I_\lambda^\infty$ satisfies $I_\lambda^\infty(u)\geq\delta$.
\end{lemma}

\begin{proof}
Letting $u$ be a nontrivial critical point of $I_\lambda^\infty$, from the
 Pohoz\u{a}ev identity \eqref{pohozaev identity}, we have
\[
I_\lambda^\infty(u)=\frac{s}N\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2.
\]
Since  (A20) and (A21) hold, for any $\varepsilon>0$, there exists $C(\varepsilon)>0$ such that
\[
\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2+V|u|^2
\leq \varepsilon\int_{\mathbb{R}^N}|u|^2+C(\varepsilon)\int_{\mathbb{R}^N}|u|^{2_s^*}.
\]
Thus, $\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2\leq C\int_{\mathbb{R}^N}|u|^{2_s^*}$.
On the other hand, by the Sobolev embedding theorem, we have 
$\int_{\mathbb{R}^N}|u|^{2_s^*}\leq\tilde C (\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2)^{\frac{2_s*}2}$.
Since $u\neq 0$, there exists a constant $\delta_0 >0$ such that 
$\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u|^2\geq \delta_0$ and so
$I_\lambda^\infty(u)\geq \delta:=s\delta_0/N$. The proof is complete.
\end{proof}

Now, we  give the decomposition of a BPS sequence.

\begin{proposition}  \label{proposition1}
 Assume {\rm (A20)--(A25)} hold. If $\max\{2,2_s^*-2\}<p<2_s^*$,  
for almost every $\lambda\in [1/2,1]$, $\{u_n\}$ given in Lemma \ref{Lemma2.2}
is the BPS sequence at the MP value $c_\lambda$. Moreover,
$c_\lambda<\frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}$.
Then there exist a subsequence, still denoted by $\{u_n\}$, an integer 
$k\in \mathbb{N}\cup\{0\}$ and $v_\lambda^j\in H^s(\mathbb{R}^N)$ for $1\leq j\leq k$, such that
\begin{itemize}
\item [(i)] $u_n\to u_\lambda$ weakly in $H^s(\mathbb{R}^N)$ and $I_\lambda'(u_\lambda)=0$,
\item [(ii)] $v_\lambda^j\neq0,v_\lambda^j\geq 0$ and $I_\lambda^{\infty'}(v_\lambda^j)=0$ 
 for $1\leq j\leq k$,
\item [(iii)] $c_\lambda=I_\lambda(u_\lambda)+\sum_{j=1}^k I_\lambda^\infty(v_\lambda^j)$,
\item [(iv)] $\|u_n-u_0-\sum_{j=1}^k v_\lambda^j(\cdot-y_n^j)\|\to 0$.
\end{itemize}
where  $|y_n^j|\to\infty$ and $|y_n^i-y_n^j|\to \infty$ as $n\to \infty$ for any $i\neq j$.
\end{proposition}

\begin{proof}  
For $\lambda\in [1/2,1]$, let $\{u_n\}\subset H^s(\mathbb{R}^N),u_n\geq0$ be given in
Lemma \ref{Lemma2.2}. Since $\{u_n\}$ is bounded, there exist a subsequence
denoted by $\{u_n\}$ and $u_\lambda\in H^s(\mathbb{R}^N)$ satisfying $u_n\to u_\lambda$
weakly  in $H^s(\mathbb{R}^N)$ and $u_n\to u_\lambda$ a.e. in $\mathbb{R}^N$. 
It is not hard to  verify that $I_\lambda'(u_\lambda)=0$.
\smallskip

\noindent\textbf{Step 1.} 
Let $v_n^1=u_n-u_\lambda$. If $v_n^1\to 0$ strongly in $H^s(\mathbb{R}^N)$, 
the Proposition holds with $k=0$.

\noindent\textbf{Step 2.} We  claim that if $v_n^1\not\to 0$ strongly, 
then $\lim_{n\to \infty}\sup_{z\in \mathbb{R}^N}\int_{B_1(z)}|v_n^1|^2>0$.
Since $I_\lambda(u_n)\to c_\lambda$, by Lemma \ref{Brezis-Lieb lemma}, we have
\begin{equation} \label{guji1}
c_\lambda-I_\lambda(u_\lambda)=I_\lambda(v_n^1)+o(1).
\end{equation}
Since $v_n^1\rightharpoonup 0$, by (A24) and Lemma \ref{lemma1.1}, we have
\begin{align*}
I_\lambda^\infty(v_n^1)-I_\lambda(v_n^1)
&=\int_{\mathbb{R}^N}(V(\infty)-V(x))|v_n^1|^2\\
&=\int_{B_{R}(0)}(V(\infty)-V(x))|v_n^1|^2
 +\int_{\mathbb{R}^N\backslash B_{R}(0)}(V(\infty)-V(x))|v_n^1|^2\\
&\to 0.
\end{align*}
Consequently,
\begin{equation} \label{guji2}
c_\lambda-I_\lambda(u_\lambda)=I_\lambda^\infty(v_n^1)+o(1).
\end{equation}
Suppose $\lim_{n\to \infty}\sup_{z\in \mathbb{R}^N}\int_{B_1(z)}|v_n^1|^2=0$.
By  Lemma \ref{lionslemma}, we have
\begin{equation} \label{lions1}
v_n^1\to 0 \quad \text{in }  L^t(\mathbb{R}^N), \quad \forall t\in(2,2_s^*).
\end{equation}
Let $f(t)=h(t)+(t^+)^{2_s^*-1}$, from (A20) and (A21), for any $\varepsilon>0$, 
there exists $C(\varepsilon)>0$ such that
\[
\big|\int_{\mathbb{R}^N}H(v_n^1)\big|
\leq \varepsilon\Big(\int_{\mathbb{R}^N}|v_n^1|^2+|v_n^1|^{2_s^*}\Big)+C(\varepsilon)\int_{\mathbb{R}^N}|v_n^1|^r,
\]
where $r<2_s^*$. Since $v_n^1\in H^s(\mathbb{R}^N)$,  from \eqref{lions1}, we obtain
\[
\big|\int_{\mathbb{R}^N}H(v_n^1)\big|\leq \varepsilon C+o(1),
\]
which implies $\int_{\mathbb{R}^N}H(v_n^1)=o(1)$ since $\varepsilon$ is small enough.
Furthermore, by the Brezis-Lieb lemma, we have
\[
\int_{\mathbb{R}^N}|v_n^1|^{2_s^*}
=\int_{\mathbb{R}^N}|u_n|^{2_s^*}-\int_{\mathbb{R}^N}|u_\lambda|^{2_s^*}+o(1).
\]
Thus, \eqref{guji1} reduces to
\begin{equation} \label{guji3}
c_\lambda-I_\lambda(u_\lambda)=\frac{1}2\|v_n^1\|^2-\frac{\lambda}{2_s^*}
\int_{\mathbb{R}^N}|v_n^1|^{2_s^*}+o(1).
\end{equation}
Noting that  $I'_\lambda(u_n)v_n^1\to 0$ and $I'_\lambda(u_\lambda)v_n^1=0$,
by direct calculation, we obtain
\[
\|v_n^1\|^2-\lambda\int_{\mathbb{R}^N}\left(f(u_n)-f(u_\lambda)\right)v_n^1
=I'_\lambda(u_n)v_n^1-I'_\lambda(u_\lambda)v_n^1\to 0.
\]
By Lemma \ref{splittinglemma},
\begin{align*}
\int_{\mathbb{R}^N}(f(u_n)-f(u))v_n^1
&=\int_{\mathbb{R}^N}f(v_n^1)v_n^1+o(1)\|v_n^1\|\\
&=\int_{\mathbb{R}^N}h(v_n^1)v_n^1+\int_{\mathbb{R}^N}|v_n^1|^{2_s^*}+o(1)\|v_n^1\|.
\end{align*}
By \eqref{lions1} and similar argument as above, we have
\[
\int_{\mathbb{R}^N}\left(f(u_n)-f(u_\lambda)\right)v_n^1
=\int_{\mathbb{R}^N}|v_n^1|^{2_s^*}+o(1). 
\]
Therefore,
\begin{equation} \label{guji4}
\|v_n^1\|^2-\lambda\int_{\mathbb{R}^N}|v_n^1|^{2_s^*}=o(1).
\end{equation}
Combining \eqref{guji3} with \eqref{guji4}, we obtain
 $c_\lambda-I_\lambda(u_\lambda)=\frac{s}N\|v_n^1\|^2+o(1)$.

Noting that $I_\lambda'(u_\lambda)=0$, from Pohoz\v{a}ev identity 
\eqref{pohozaev identity} and Sobolev embedding theorem, we obtain
\begin{align*}
& I_\lambda(u_\lambda) \\
&=\frac{s}N\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2-\frac{1}{2N}
 \int_{\mathbb{R}^N}\langle\nabla V(x),x\rangle u_\lambda^2\\
&\geq \frac{s}N\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2-\frac{1}{2NS_s}
 \|\max\{\langle\nabla V(x),x\rangle,0\}\|_{L^{\frac{N}{2s}}}
 \int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2.
\end{align*}
Condition (A25) implies $I_\lambda(u_\lambda)\geq 0$.
Thus $c_\lambda-I_\lambda(u_\lambda)<\frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}$.
 On the other hand, since $v_n^1\not\to 0$ strongly,  there exists a
constant $l>0$ such that $\|v_n^1\|^2\to l$. 
Set $\|(-\Delta)^{s/2}v_n^1\|_{L^2}^2=\tilde l<l$, then
\[
S_s=\inf_{u\in H^s(\mathbb{R}^N),{u\neq 0}}
\frac{\|(-\Delta)^{s/2}u\|^2_{L^2}}{\|u\|^2_{L^{2_s^*}}}
\leq \frac{\tilde l}{(\frac{l}\lambda)^{\frac{2}{2_s^*}}}
\leq l^{\frac{2s}{N}}\lambda^{\frac{N-2s}N}.
\]
So we have $l\geq \frac{S_s^{\frac{N}{2s}}}{\lambda^{\frac{N-2s}{2s}}}$. 
Consequently, $c_\lambda-I_\lambda(u_\lambda)
\geq\frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}$, which is a contradiction. 
The claim is true.
\smallskip

\noindent\textbf{Step 3.} From the argument in step 2, 
if $v_n\rightharpoonup 0$, then  
\[
\lim_{n\to \infty}\sup_{z\in \mathbb{R}^N}\int_{B_1(z)}|v_n^1|^2>0.
\]
Thus, after extracting a subsequence if necessary, there exist 
$\{z_n^1\}\subset \mathbb{R}^N$ and  $v_\lambda^1\in H^s(\mathbb{R}^N)$ 
such that $|z_n^1|\to\infty$ and
\begin{itemize}
\item[(i)] $\lim_{n\to \infty}\int_{B_1(z_n^1)}|v_n^1|^2>0$, 
\item[(ii)] $v_n^1(\cdot+z_n^1)\rightharpoonup v_\lambda^1\neq 0$,
\item[(iii)] $I_\lambda^{\infty'} (v_\lambda^1)=0$.
\end{itemize}
Clearly (i), (ii) are standard and the point is to show (iii). 
Set $u_n^1=v_n^1(\cdot+z_n^1)$. To prove $I_\lambda^{\infty'} (v_\lambda^1)=0$, 
it suffices to prove  $I_\lambda^{\infty'} (u_n^1)\to 0$. For any 
$\varphi\in C_0^\infty(\mathbb{R}^N)$, from $I_\lambda'(v_n^1)\to 0$, we have
\begin{align*}
&I_\lambda'(v_n^1)\varphi(\cdot -z_n^1) \\
&=\int_{\mathbb{R}^N}(-\Delta)^{s/2}v_n^1(x+z_n^1)(-\Delta)^{s/2}\varphi(x)dx
 +\int_{\mathbb{R}^N}V(x+z_n^1)v_n^1(x+z_n^1)\varphi(x)dx\\
&\quad -\int_{\mathbb{R}^N}f(v_n^1(x+z_n^1))\varphi(x)dx\\
&=\int_{\mathbb{R}^N}(-\Delta)^{s/2}u_n^1(x)(-\Delta)^{s/2}\varphi(x)dx
+\int_{\mathbb{R}^N}V(x+z_n^1)u_n^1(x)\varphi(x)dx\\
&\quad -\int_{\mathbb{R}^N}f(u_n^1(x))\varphi(x)dx
\to 0.
\end{align*}
Since $|z_n^1|\to\infty$ and $\varphi\in C_0^\infty(\mathbb{R}^N)$, by (A24), we obtain
\[
\int_{\mathbb{R}^N}V(x+z_n^1)u_n^1(x)\varphi(x)dx
 \to \int_{\mathbb{R}^N}V(\infty)u_n^1(x)\varphi(x)dx.
\]
Thus,
\begin{align*}
I_\lambda^{\infty'}(u_n^1)
&=\int_{\mathbb{R}^N}(-\Delta)^{s/2}u_n^1(x)(-\Delta)^{s/2}\varphi(x)dx
 +\int_{\mathbb{R}^N}V(\infty)u_n^1(x)\varphi(x)dx\\
&\quad -\int_{\mathbb{R}^N}f(u_n^1(x))\varphi(x)dx\to 0.
\end{align*}
Then we obtain $I_\lambda^{\infty'} (v_\lambda^1)=0$ since $u_n^1\rightharpoonup v_\lambda^1$.
On the other hand, from \eqref{guji2}, it is easy to see that
 $c_\lambda-I_\lambda(u_\lambda)=I_\lambda^\infty(u_n^1)+o(1)$.

So, we obtain a bounded sequence $\{u_n^1\}$ with $u_n^1\rightharpoonup v_\lambda^1\neq 0$ satisfying
\[
I_\lambda^\infty(u_n^1)\to c_\lambda-I_\lambda(u_\lambda),\quad 
I_\lambda^{\infty'}(u_n^1)\to 0,\quad I_\lambda^{\infty'} (v_\lambda^1)=0.
\]
Let $v_n^2=u_n^1-v_\lambda^1$. Then $u_n=u_\lambda+v_\lambda^1(\cdot-z_n^1)+v_n^2(\cdot-z_n^1)$. 
If $v_n^2\to 0$ strongly in $H^s(\mathbb{R}^N)$, we have
\begin{gather*}
c_\lambda-I_\lambda(u_\lambda)=I_\lambda^\infty(v_\lambda^1),\\
\|u_n-u_\lambda-v_\lambda^1(\cdot-z_n^1)\|\to 0.
\end{gather*}
If $v_n^2\not\to 0$ strongly, similarly as \eqref{guji1} and \eqref{guji2},
 we have
\[
c_\lambda-I_\lambda(u_\lambda)-I_\lambda^\infty(v_\lambda^1)=I_\lambda^\infty(v_n^2)+o(1),I_\lambda^{\infty'}(v_n^2)\to 0.
\]
By the same argument as step 2, we obtain 
$\lim_{n\to \infty}\sup_{z\in \mathbb{R}^N}\int_{B_1(z)}|v_n^2|^2>0$. 
Then, there exist $\{z_n^2\}\subset \mathbb{R}^N$ and  $v_\lambda^2\neq 0$ 
such that $|z_n^2|\to\infty$ and
\begin{itemize}
\item[(i)] $\lim_{n\to \infty}\int_{B_1(z_n^2)}|v_n^1|^2>0$, 
\item[(ii)] $ v_n^2(\cdot+z_n^2)\rightharpoonup v_\lambda^2$,
\item[(iii)] $I_\lambda^{\infty'} (v_\lambda^2)=0$.
\end{itemize}
Set $u_n^2=v_n^2(\cdot+z_n^2)$. Then, $\{u_n^2\}$ is a bounded sequence
 satisfying $u_n^2\rightharpoonup v_\lambda^2$ and
\[
I_\lambda^\infty(u_n^2)\to c_\lambda-I_\lambda(u_\lambda)-I_\lambda^\infty(v_\lambda^1),\quad
I_\lambda^{\infty'}(u_n^2)\to 0.
\]
Let $v_n^3=u_n^2-v_\lambda^2$. Then 
$u_n=u_\lambda+v_\lambda^1(\cdot-z_n^1)+v_\lambda^2(\cdot-z_n^1-z_n^2)+v_n^3(\cdot-z_n^1-z_n^2)$. 
If $v_n^3\to 0$ strongly in $H^s(\mathbb{R}^N)$, we have
\begin{gather*}
c_\lambda=I_\lambda(u_\lambda)+I_\lambda^\infty(v_\lambda^1)+I_\lambda^\infty(v_\lambda^2),\\
\|u_n-u_\lambda-v_\lambda^1(\cdot-z_n^1)-v_\lambda^2(\cdot-z_n^1-z_n^2)\|\to 0.
\end{gather*}
Otherwise, we repeat the procedure above. From Lemma \ref{energyest}, 
we can terminate our arguments by repeating the above proof by finite $k$ steps. 
That is, let $y_n^j=\sum_{i=1}^{j}z_n^i$, then
\begin{gather*}
c_\lambda=I_\lambda(u_\lambda)+\sum_{j=1}^{k}I_\lambda^\infty(v_\lambda^j),\\
\|u_n-u_\lambda-\sum_{j=1}^{k}v_\lambda^j(\cdot-y_n^j)\|\to 0.
\end{gather*}

\noindent\textbf{Step 4.} Now, we show that after extracting a subsequence of 
$\{y_n^j\}$ and redefining $\{v_\lambda^j\}$ if necessary, (iii), (iv)  hold for 
$|y_n^j|\to\infty$ and $|y_n^i-y_n^j|\to \infty$ as $n\to \infty$ for any $i\neq j$.
Let $A=\{1,2,\cdot\cdot\cdot,k\}$.  From 
$u_n-u_\lambda-\sum_{j=1}^{k}v_\lambda^j(\cdot-y_n^j)\to 0$ and 
$u_n\to u_\lambda$ a.e. in $\mathbb{R}^N$, we obtain that
$\sum_{j=1}^{k}v_\lambda^j(\cdot-y_n^j)\to 0$ a.e. in $\mathbb{R}^N$. 
Since $v_\lambda^j\geq0$ for any $j$, it follows that $|y_n^j|\to \infty$. 
For $y_n^i$, assume $A_i=\{y_n^j:|y_n^i-y_n^j| \text{ is bounded for }n \}$, 
then up to a sequence, there exists some $\tilde v_\lambda^i\in H^s(\mathbb{R}^N)$ 
such that $\sum_{j\in A_i}v_\lambda^j(\cdot+y_n^i-y_n^j)\to \tilde v_\lambda^i$ 
strongly in $H^s(\mathbb{R}^N)$. Then 
$\|u_n-u_\lambda-\tilde v_\lambda^i(\cdot-y_n^i)-\sum_{j\in (A\setminus A_i)}v_\lambda^j(\cdot-y_n^j)
\|\to 0$. Since $v_\lambda^j(j\in A)$ is the critical point of $I_\lambda^{\infty'}$, we have 
$I_\lambda^{\infty'}(\tilde v_\lambda^i)=0$. Then we redefine 
$v_\lambda^i:=\tilde v_\lambda^i$, and then 
$\|u_n-u_\lambda-\sum_{j\in (A\setminus A_i)\cup \{i\}}v_\lambda^j(\cdot-y_n^j)\|\to 0$ 
holds as $n\to \infty$. By repeating the argument above at most $(k-1)$ times and 
redefining $\{v_\lambda^j\}$ if necessary, there exists $\Lambda\subset A$ such that
\begin{gather*}
|y_n^j|\to\infty, \quad |y_n^i-y_n^j|\to \infty,\quad \forall i\neq j,\quad  n\to \infty, \\
\|u_n-u_\lambda-\sum_{j\in \Lambda}v_\lambda^j(\cdot-y_n^j)\|\to 0.
\end{gather*}
The proof is complete.
\end{proof}

If $V(x)\equiv V>0$, we can get the similar decomposition of the BPS sequence 
for the autonomous problem \eqref{p1}. Denote the energy functional of 
autonomous problem \eqref{p1} and auxiliary energy functional by $J$ 
and $J_\lambda$ $(\lambda\in [1/2,1])$ respectively. Let $c_\lambda$ be the MP value 
for $J_\lambda$, then we have the following result.


\begin{corollary} \label{corollary} 
Assume $V(x)\equiv V>0$ and {\rm (A20)--(A22)} hold. For $\lambda\in[1/2,1]$, 
if $\{u_n\}\subset H^s(\mathbb{R}^N)$ is a sequence such that 
$u_n\geq0,\|u_n\|<\infty, J_\lambda(u_n)\to c_\lambda$ and $J_\lambda'(u_n)\to 0$,
furthermore $c_\lambda<\frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}$. 
Then there exist a subsequence of $\{u_n\}$, an integer $l\in \mathbb{N}\cup\{0\}$ 
and $w_\lambda^j\in H^s(\mathbb{R}^N)$ for $1\leq j\leq l$ such that
\begin{itemize}
\item [(i)] $u_n\to u_\lambda$ weakly in $H^s(\mathbb{R}^N)$ with $J_\lambda'(u_\lambda)=0$,
\item [(ii)] $w_\lambda^j\neq0,w_\lambda^j\geq 0$ and $J_\lambda'(w_\lambda^j)=0$ for $1\leq j\leq l$,
\item [(iii)] $c_\lambda=J_\lambda(u_\lambda)+\sum_{j=1}^l J_\lambda(w_\lambda^j)$,
\item [(iv)] $\|u_n-u_0-\sum_{j=1}^l w_\lambda^j(\cdot-y_n^j)\|\to 0$,
\end{itemize}
where  $|y_n^j|\to\infty$ and $|y_n^i-y_n^j|\to \infty$ as $n\to \infty$ for any $i\neq j$.
\end{corollary}

The proof of the above corollary is similar to Proposition \ref{proposition1}, 
we omit it here.
Now, we complete the proof of the existence of solutions of the auxiliary 
problems \eqref{auxiliaryproblem}.

\begin{lemma} \label{Lemma2.8}
Assume {\rm (A20)--(A25)} hold. If $\max\{2,2_s^*-2\}<p<2_s^*$, then
for almost every $\lambda \in [1/2,1]$, $I_\lambda$ has a positive critical point 
$u_\lambda$ satisfying $\|u_\lambda\|\geq \delta$ where $\delta>0$ independent of $\lambda$.
\end{lemma}

\begin{proof}  
From Lemmas \ref {Lemma2.2} and  \ref{Lemma2.3}, there exists a bounded sequence
 $\{u_n\}\subset H^s(\mathbb{R}^N)$, $u_n\geq0$ and 
$0<c_\lambda<\frac{s}{N\lambda^{\frac{N-2s}{2s}}}S_s^{\frac{N}{2s}}$, such that
\[
I_\lambda(u_n)\to c_\lambda, \quad I'_\lambda(u_n)\to 0.
\]
Then $u_n\to u_\lambda\geq 0$ weakly in $H^s(\mathbb{R}^N)$. 
It is obvious that $u_\lambda$ is a critical point of $I_\lambda$.

 Now, we claim $u_\lambda\neq 0$. If $u_\lambda=0$, from Proposition \ref{proposition1}, 
we can deduce that  $k>0$ since $c>0$, and 
\begin{equation} \label{clambda1}
c_\lambda=\sum_{j=1}^k I_\lambda^\infty(v_\lambda^j)\geq m_\lambda^\infty:=\inf\{I_\lambda^\infty(u):
u\in H^s(\mathbb{R}^N),u\neq 0,I_\lambda^{\infty'}(u)=0\},
\end{equation}
where $I_\lambda^{\infty'}(v_\lambda^j)=0(j=1,2,\dots,k)$. On the other hand, we infer that
\begin{equation} \label{clamba2}
c_\lambda<m_\lambda^\infty,
\end{equation}
which is contradictory to \eqref{clambda1} and then the claim is true.

From Lemma \ref{Lemma2.6}, we let $v_\lambda$  be the least energy solution of
\[
(-\Delta)^su+V(\infty)u=\lambda f(u).
\]
By Lemma \ref{Lemma2.5}, there exists $\gamma_\lambda(t)$ satisfying
$\gamma_\lambda(0)=0,I_\lambda^\infty(\gamma_\lambda(1))<0,v_\lambda\in \gamma_\lambda[0,1]$ and
\[
\max_{t\in[0,1]}I_\lambda^\infty(\gamma_\lambda(t))=I_\lambda^\infty(v_\lambda)= m_\lambda^\infty.
\]
By (A24), we have
\[
I_\lambda(\gamma_\lambda(t))<I_\lambda^\infty(\gamma_\lambda(t)), \quad \forall t\in [0,1],
\]
and  from the definition of $c_\lambda$ it follows that
\[
c_\lambda\leq \max_{t\in[0,1]}I_\lambda(\gamma_\lambda(t))<\max_{t\in[0,1]}I_\lambda^\infty(\gamma_\lambda(t))= m_\lambda^\infty.
\]

Since (A20) and (A21) hold, by the same argument as that in Lemma \ref{energyest}, 
there exists a constant $\delta_0 >0$ independent of $\lambda$ such that 
$\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2\geq \delta_0$ since
 $u_\lambda\neq 0$. Thus, there exists a $\delta>0$ independent of $\lambda$ such that 
$\|u_\lambda\|\geq \delta$. The proof is complete.
\end{proof}


\section{Proof of Theorem \ref{thm1.1}}

Lemma \ref{Lemma2.8} shows that for almost every $\lambda\in [1/2,1]$,
$I_{\lambda}(u)$ has a positive critical point $u_{\lambda}$. Thus we obtain a critical
point sequence $\{u_{\lambda}\}$ satisfying $I'_{\lambda}(u_{\lambda})=0$. 
In the following, we first show that $\{u_\lambda\}$ is a BPS sequence of $I$ 
 and then prove the convergence of $\{u_\lambda\}$ as $\lambda\to 1$. 
By analyzing the properties of  minimizing sequence, we complete the proof 
of the existence of ground state solutions of \eqref{p1}.
First, we show  the uniform boundedness of $\{u_\lambda\}$.

\begin{proposition}  \label{proposition2} 
Assume {\rm (A20)--(A25)} hold. If $\max\{2,2_s^*-2\}<p<2_s^*$, then 
$\{u_{\lambda}\}$ is bounded uniformly and there exists $\delta>0$ independent 
of $\lambda$ such that $I_\lambda(u_\lambda)\geq \delta$.
\end{proposition}

\begin{proof}
Since $u_\lambda$ is the critical point of $I_\lambda(u)$, from the Pohoz\v{a}ev
 identity \eqref{pohozaev identity}, we have
\begin{equation} \label{ilambdaest}
I_\lambda(u_\lambda)=\frac{s}N\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2
-\frac{1}{2N}\int_{\mathbb{R}^N}\langle\nabla V(x),x\rangle |u_\lambda|^2.
\end{equation}
From Proposition \ref{proposition1}, $I_\lambda(u_\lambda)\leq c_\lambda\leq c_{1/2}$
for any $\lambda\in [1/2,1]$. By the H\"older inequality and Sobolev embedding theorem,
\begin{align*}
\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2
&=\frac{N}s I_\lambda(u_\lambda)+\frac{1}{2s}\int_{\mathbb{R}^N}\langle\nabla V(x),x\rangle |u_\lambda|^2\\
&\leq \frac{N}s c_{\frac{1}2}+\frac{1}{2sS_s}\|
\max\{\langle\nabla V(x),x\rangle,0\}\|_{L^{\frac{N}{2s}}}
\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2.
\end{align*}
Condition (A25) implies that $\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2$ is
bounded uniformly independent of $\lambda$. Next, we show that $\|u_\lambda\|_{L^2}$
is bounded uniformly independent of $\lambda$.
From $I'_\lambda(u_\lambda)u_\lambda=0$, we have
$\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2+\int_{\mathbb{R}^N}V(x)|u_\lambda|^2
=\lambda\int_{\mathbb{R}^N}f(u_\lambda)u_\lambda$. Then, by (A20) and (A21),
\begin{align*}
V_0\int_{\mathbb{R}^N}|u_\lambda|^2
&\leq\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2+\int_{\mathbb{R}^N}V(x)|u_\lambda|^2\\
&\leq \lambda\varepsilon\int_{\mathbb{R}^N}|u_\lambda|^2+\lambda C(\varepsilon)\int_{\mathbb{R}^N}|u_\lambda|^{2_s^*}\\
&\leq \varepsilon\int_{\mathbb{R}^N}|u_\lambda|^2+C(\varepsilon)
\Big|\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2\Big|^{2_s^*/2}.
\end{align*}
Therefore, $\|u_\lambda\|_{L^2}$  is bounded uniformly. Now, we prove that
$I_\lambda(u_\lambda)\geq \delta>0$. From Lemma \ref{Lemma2.8}, there exists
 $\delta_0>0$ independent of $\lambda$ such that $\|u_\lambda\|\geq \delta_0$.
On the other hand,
\begin{align*}
&I_\lambda(u_\lambda) \\
&\geq \frac{s}N\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2
 -\frac{1}{2N}\int_{\mathbb{R}^N}\max\{\langle\nabla V(x),x\rangle,0\} |u_\lambda|^2\\
&\geq \frac{s}N\int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2
 -\frac{1}{2NS_s}\|\max\{\langle\nabla V(x),x\rangle,0\}\|_{L^{\frac{N}{2s}}}
 \int_{\mathbb{R}^N}|(-\Delta)^{s/2}u_\lambda|^2.
\end{align*}
Condition (A25) implies that there exists $\delta>0$ independent of $\lambda$
such that
\begin{equation} \label{ilambda1}
I_\lambda(u_\lambda)\geq \delta
\end{equation}
The proof is complete.
\end{proof}

In the following, we denote $u_\lambda$ by $u_{\lambda_j}$ and let $\lambda_j\to 1$ 
as $j\to \infty$.

\begin{lemma} \label{Lemma3.1}
Assume {\rm (A20)--(A25)} hold, if $\max\{2,2_s^*-2\}<p<2_s^*$, then 
the sequence $\{u_{\lambda_j}\}$ is a BPS sequence for $I$ satisfying 
$\limsup_{j\to \infty}I(u_{\lambda_j})\leq c_1$ and $\|u_{\lambda_j}\|\not\to 0$.
\end{lemma}

\begin{proof}
From Lemma \ref{Lemma2.8}, we have $\|u_{\lambda_j}\|\not\to 0$.
 It follows from Proposition \ref{proposition2} that $\|u_{\lambda_j}\|$ 
is bounded uniformly, and consequently $\int_{\mathbb{R}^N} F(u_{\lambda_j})$ 
is bounded by (A20) and (A21).  Property (iii) in Proposition \ref{proposition1} 
shows that $I_{\lambda_j}(u_{\lambda_j})\leq c_{\lambda_j}$ for any $u_{\lambda_j}$.
Thus, from
\begin{equation} \label{iest}
I(u_{\lambda_j})=I_{\lambda_j}(u_{\lambda_j})+(\lambda_j-1)\int_{\mathbb{R}^N} F(u_{\lambda_j}),
\end{equation}
we obtain $\limsup_{j\to \infty}I(u_{\lambda_j})\leq c_1$ and $I'(u_{\lambda_j})\to 0$.
\end{proof}

\begin{proof}[Completion of the proof of the Theorem \ref{thm1.1}]
From Lemma \ref{Lemma3.1}, inequality \eqref{ilambda1},\break and \eqref{iest}, 
there exists a subsequence still denoted by $\{u_{\lambda_j}\}$ satisfying
\begin{itemize}
\item[(i)] $\{u_{\lambda_j}\}$  is bounded,
\item[(ii)] $I(u_{\lambda_j})\to c\leq c_1$,
\item[(iii)] $I'(u_{\lambda_j})\to 0$,
\end{itemize}
where $c>0$. That is to say, there exists a BPS sequence $\{u_{\lambda_j}\}$ 
satisfying the assumptions of Lemma \ref{Lemma2.8} for $\lambda=1$. 
Thus, there exists a nontrivial critical point $u_0$ for $I$ satisfying 
$I(u_0)\leq c_1$.

Next, we show the existence of a ground state solution. Let
\[
m=\inf\{I(u):u\in H^s(\mathbb{R}^N),u\neq 0,I'(u)=0\}.
\]
Obviously, $m\leq I(u_0)\leq c_1=\frac{s}NS_s^{\frac{N}{2s}}$. 
Set $\{u_n\}$ be a sequence of nontrivial critical points of $I$ satisfying 
$I(u_n)\to m$.  Since $I(u_n)$ is bounded, similar proof as that in 
Proposition \ref{proposition2} for $\lambda=1$, we obtain that $\{u_n\}$ is
bounded uniformly and there exists $\delta>0$ such that $I(u_n)\geq \delta>0$. 
Thus $m>0$. So, $\{u_n\}$ is a BPS sequence
satisfying the following conditions,
\begin{itemize}
\item[(i)] $\{u_n\}$ is bounded,
\item[(ii)] $I(u_n)\to m\leq c_1$,
\item[(iii)] $I'(u_n)=0$,
\end{itemize}
 From Proposition \ref{proposition1}, there exists $\tilde u$ such that 
$I'(\tilde u)=0$ and $I(\tilde u)\leq m$.

Now, we claim $\tilde u\neq 0$.
Otherwise, $\tilde u=0$. Then, by Proposition \ref{proposition1}, we have
\[
m=\sum_{j=1}^k I^\infty(w^j)\geq m^\infty
:=\inf\{I^\infty(u):u\in H^s(\mathbb{R}^N),u\neq 0,I^{\infty'}(u)=0\}
\]
for $k>0$ and $w^j(j=1,2,\dots,k)$ are the critical points of $I^\infty$.
On the other hand, similar argument as that in Lemma \ref{Lemma2.8}, 
there exists  $\gamma(t)$ such that
\[
\max_{t\in[0,1]}I^\infty(\gamma(t))= m^\infty.
\]
From the definition of $c_1$, we obtain $m\leq c_1 \leq \max_{t\in[0,1]}I(\gamma(t))$. 
By (A24), we obtain
\[
m\leq c_1< m^\infty,
\]
which is a contradiction. Thus, the claim is true.
Then $I(\tilde u)\geq m$ since $I'(\tilde u)=0$ and $\tilde u\neq 0$. 
So, there exists a critical point $\tilde u\neq 0$ such that $I(\tilde u)=m$. 
The proof is complete.
\end{proof}

\subsection*{Acknowledgements}
This work is supported by the National Natural Science Foundation 
of China (11271364).

\begin{thebibliography}{00}

\bibitem{G.Alberti} G. Alberti, G. Bouchitt\'{e}, P. Seppecher;
\emph{Phase transition with the line-tenstion effect}.
 Arch. Ration. Mech. Anal., 144 (1998), 1-46.

\bibitem{Alvesfrac} C. O. Alves, O. H. Miyagaki;
\emph{Existence and concentration of solution for a class of
fractional elliptic equation in $\mathbb{R}^N$ via penalization method}, 
Calc. Var. Partial Differential Equations. 55 (2016), 47.

\bibitem{B.Barriosa} B. Barriosa, E. Coloradoc, R. Servadeid, F. Soriaa;
\emph{A critical fractional equation with concave-convex power nonlinearities}, 
Ann. I. H. Poincar\'{e}-AN., 32 (2015), 875-900.


\bibitem{Barriosa1} B. Barriosa, E. Colorado, A. de Pablo, U. S\'anchez;
\emph{On some critical problems for the fractional Laplacian operator}, 
J. Differential Equations, 252 (2012), 6133-6162.


\bibitem{Berestycki} H. Berestycki, P. Lions;
\emph{Nonlinear scalar field equations I. Existence of a ground state},
 Arch. Ration. Mech. Anal., 82 (1990), 90-117.

\bibitem{bisci} G. M. Bisci, V. Radulescu;
\emph{Ground state solutions of scalar field fractional Schr\"odinger equation},
 Calc. Var. Partial Differential Equatioans, 54(2015) , 2985-3008.

\bibitem{Cabre} X. Cabr\'e, J. Tan;
\emph{Positive solutions of nonlinear problems involving the square root 
of the Laplacian}, Adv. Math., 224 (2010), 2052-2093.

\bibitem{L.A.Caffarelli} L. A. Caffarelli, S. Salsa, L. Silvestre;
\emph{Regularity estimates for the solution and the free boundary of 
the obstacle problem for the fractional Laplacian},
 Invent. Math.,  171(2) (2008), 425-461.

\bibitem{Caffarelli} L. Caffarelli, L. Silvestre;
\emph{An extension problem related to the fractional Laplacian}, 
Commun. PDEs, 32 (2007), 1245-1260.

\bibitem{changxiaojun} X. J. Chang, Z. Q. Wang;
\emph{Nodal and multiple solutions of nonlinear problems involving the 
fractional Laplacian}. J. Differential Equations, 256 (2014), 2965-2992.

\bibitem{changxiaojun2} X. J. Chang, Z. Q. Wang;
\emph{Ground state of scalar field equations involving a fractional 
Laplacian with general nonlinearity}, Nonlinearity, 26 (2013), 479-494.

\bibitem{chenzheng}  G. Chen, Y. Zheng;
\emph{Concentration phenomena for fractional nonlinear Schr\"odinger equations},
Commun. Pure Appl. Anal., 13 (2014), 2359-2376.

\bibitem{A.Cotsiolis} A. Cotsiolis, N. K. Tavoularis;
\emph{Best constants for Sobolev inequalities for higher order fractional
 derivatives}, J. Math. Anal. Appl., 295 (2004), 225-236.

\bibitem{Davila} J. D\'{a}vila, M. del Pino, J. Wei;
\emph{Concentrating standing waves for the fractional nonlinear Schr\"oding
equation}, J. Differ. Equ., 256 (2) (2014), 858-892.

\bibitem{FMM}  M. M. Fall,  F. Mahmoudi, E. Valdinoci;
\emph{Ground states and concentration phenomena for the fractional 
Schr\"odinger equations}, Nonlinearity, 28 (2015), 1937-1961.


\bibitem{Q.Y.Guan} Q. Y. Guan, Z. M. Ma;
\emph{Boundary problems for fractional Laplacians},
 Stoch. Dyn., 593 (2005), 385-424.

\bibitem{xiaominghe} X. M. He, W. M. Zou;
\emph{Existence and concentration result for the fractional 
Schr\"odinger equations with critical nonlinearities},
Calc. Var. Partial Differential Equations, 55(4) (2016) :91.

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

\bibitem{L.JeanjeanandKazunagaTanaka} L. Jeanjean, K. Tanaka;
\emph{A Positive solution for a nonlinear Schr\"odinger equation on
$\mathbb{R}^N$}, Indiana. Univ. Math. J., 54 (2005), 443-464.

\bibitem{JeanTa} L. Jeanjean, K. Tanaka;
\emph{A remark on least energy solutions in $\mathbb {R^N}$}, 
Proc. Amer. Math. Soc., 13 (2002), 2399-2408.

\bibitem{NLaskin1} N. Laskin;
\emph{Fractional Schr\"odinger equation}, Phys. Rev. E, 66 (2002), 05618.

\bibitem{NLaskin2} N. Laskin;
\emph{Fractional quantum mechanics}, Phys. Rev. E, 62 (2000), 3135.

\bibitem{Laskin} N. Laskin;
\emph{Fractional quantum mechanics ans L\'{e}vy path integrals}, 
Phys. Lett. A,  268 (4-6) (2000), 298-305.

\bibitem{lionsembedding} P. L. Lions;
\emph{Sym\'{e}trie et compacit\'{e} dans les espaces de Sobolev}, 
J. Funct. Analysis, 49 (1982), 315-334.


\bibitem{Nezza} E. Di Nezza, G. Palatucci, E. Valdinoci;
\emph{Hitchiker's guide to the fractional Sobolev space}, 
Bull. Sci. Math., 136 (2012),  521-573.

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

\bibitem{seok} J. Seok;
\emph{Spike-layer solutions to nonlinear fractional Schr\"oding equations
with almost optimal nonlinearities},
Electron. J. Differential Equations. 2015 no. 196 (2015), 1-19.

\bibitem{shangxzhangj} X. D. Shang, J. H. Zhang;
\emph{Concentrating solutions of nonlinear fractional Schr\"odinger
equation with potentials}, J. Differ. Equ., 258(4) (2015), 1106-1128.

\bibitem{Silvestre} L. Silvestre;
\emph{Regularity of the obstacle problem for a fractional power of the 
Laplace ooperator}, Comm. Pure Apple. Math., 60(1) (2007), 67-112.

\bibitem{Y.Sire} Y. Sire, E. Valdinoci;
\emph{Fractional Laplacian phase transtion and boundary reactions: 
a geometric inequality and a symmetry result}, J. Funct. Anal., 256(6)
(2009), 1842-1864.

\bibitem{simonesecchi} S. Secchi;
\emph{On fractional Schr\"odinger equations in $\mathbb{R}^N$ without the
 Ambrosetti-Rabinowitz condition}. Topological Methods in Nonlinear Analysis.
 47(1) (2016), 19-41.

\bibitem{J.Tan} J. Tan;
\emph{The Brezis-Nirenberg type problem involving the square root of the Laplacian},
 Calc. Var., 42 (2011), 21-41.

\bibitem{tengkaiming} K. M. Teng, X. M. He;
\emph{Ground state solutions for fractional Schr\"odinger equations with
critical Sobolev exponent}, Commun. Pure Apple. Anal., 16 (2016), 991-1008.

\bibitem{weiyuanhong} Y. H. wei, X. F. Su;
\emph{Multiplicity of solutions for non-local elliptic equations driven by 
the fractional Laplacian}, Calc. Var., 52 (2015), 95-124.

\bibitem{J.Zhang} J. Zhang, W. M. Zou;
\emph{The critical case for a Berestycki-Lions theorem}, 
Science China Mathematics, 57 (3) (2014), 541-554.

\bibitem{brezis-lieb} J. J. Zhang, J. M. do O, M. Squassina;
\emph{Schr\"odinger-Poisson systems with a general critical nonlinearity}.
Commun. Contemp. Math. (published online 2016).

\bibitem{zjjjm} J. J. Zhang, J. M. do O, M. Squassina;
\emph{Fractional Schr\"odinger-Poisson systems with a general subcritical
or critical nonlinearity}, Adv. Nonlinear Stud. 16 (1) (2016), 15-30.

\bibitem{zjjzwm2} J. J. Zhang, W. M. Zou;
\emph{A Berestycki-Lions Theorem revisited}, Commun. Contemp. Math. 14 (5)
(2012),  1250033.

\bibitem{zjjzwm} J. J. Zhang, W. M. Zou;
\emph{Solutions concentrating around the saddle points of the potential 
for critical schr\"odinger equations}, Calc. Var. Partial Differential Equations.
 54 (4) (2015), 4119-4142.

\bibitem{zhangxia} X. Zhang, B. L. Zhang, D. repov\u{s};
\emph{Existence and symmetry of solutions for critical fractional Schr\"odinger
equations with bounded potentials}. Nonlinear Analysis, 142 (2016), 48-68.


\end{thebibliography}

\end{document}