\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 25, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/25\hfil Multiple solutions]
{Multiple solutions for fractional \\ Schr\"odinger equations}

\author[H. Shi, H. Chen \hfil EJDE-2015/25\hfilneg]
{Hongxia Shi, Haibo Chen}

\address{Hongxia Shi \newline
School of Mathematics and Statistics, Central South University,
Changsha, 410083 Hunan, China}
\email{shihongxia5617@163.com}

\address{Haibo Chen (corresponding author)\newline
School of Mathematics and Statistics, Central South University,
 Changsha, 410083 Hunan, China}
\email{math\_chb@163.com}

\thanks{Submitted November 16, 2014. Published January 27, 2015.}
\subjclass[2000]{35B38, 35G99}
\keywords{Fractional Schr\"odinger equations; variational methods;
\hfill\break\indent  Morse theory; local linking}

\begin{abstract}
 In this article we study the fractional Schr\"odinger equations
 $$
 (-\Delta)^{\alpha}u+V(x)u=f(x,u)  \quad\text{in }\mathbb{R}^{N},
 $$
 where $0<\alpha<1$, $N\geq2$, $(-\Delta)^{\alpha}$ stands for the
 fractional Laplacian of order $\alpha$. First by using Morse theory in
 combination with local linking arguments, we prove the existence  of at least
 two nontrivial solutions. Next we prove that the problem has $k$ distinct
 pairs of solutions by using the Clark theorem.
\end{abstract}

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

\section{Introduction and main results}

In this article, we consider the  fractional Schr\"odinger equation
\begin{equation}
(-\Delta)^{\alpha}u+V(x)u=f(x,u)  \quad\text{in }\mathbb{R}^{N},\label{e1.1}
\end{equation}
where $0<\alpha<1$, $N\geq2$, $(-\Delta)^{\alpha}$ stands for the
fractional Laplacian of order $\alpha$, $V\in C(\mathbb{R}^{N},\mathbb{R})$
and $f\in C(\mathbb{R}^{N}\times\mathbb{R},\mathbb{R})$.

When $\alpha=1$, \eqref{e1.1} becomes the classical Schr\"odinger equation
\begin{equation}
-\Delta u+V(x)u=f(x,u)  \quad\text{in }\mathbb{R}^{N}. \label{e1.2}
\end{equation}
In recent years, the existence and multiplicity of standing wave solutions
of \eqref{e1.2} have been widely studied, we refer the readers
to \cite{Li,Rabin,A,Tang2,Tang,Tang1} and the references therein.

When $0<\alpha<1$, \eqref{e1.1} is a nonlocal model known as nonlinear 
fractional Schr\"odinger equation. The nonlocal model has attracted
 much attentions recently. For the case of a bounded domain, 
Ricceri \cite{Ricceri} established a theorem tailor-made for a class of nonlocal
problems involving nonlinearities with bounded primitive. 
In \cite{B3}, Molica Bisci and Repov\v{s} studied a class of nonlocal 
fractional Laplacian equations depending on two real parameters and obtained 
the existence of three weak solutions by exploiting the result established 
by Ricceri in \cite{Ricceri}. For more related results, we refer the readers 
to \cite{B1,B2} and the references therein.

Equations of the form \eqref{e1.1} in the whole space $\mathbb{R}^{N}$ were studied 
by a number of authors.
See, for instance, \cite{X,Dipierro,Felmer,Secchi,S,Shang,Wei}
 and the references therein.
 Felmer,  Quaas and Tan \cite{Felmer} studied the existence and regularity 
of positive solution of \eqref{e1.1} with $ V(x) \equiv 1$ for
general $s\in(0,1)$ when $f$ has subcritical growth and satisfies the 
Ambrosetti-Rabinowitz((AR) for short) condition. 
Secchi \cite{Secchi} obtained the existence of ground state solutions of 
\eqref{e1.1} for general $s\in(0,1)$
when $V(x)\to +\infty$ as $|x|\to +\infty$ and (AR)  condition holds. 
In \cite{Dipierro}, the authors looked for radially symmetric solutions 
of \eqref{e1.1} when $V$ and $f$ do not depend explicitly on the space
variable $x$. In \cite{Wei}, the authors obtained the existence of infinitely
 many weak solutions for \eqref{e1.1}
by variant fountain theorem established by Zou in \cite{Zou} when $f$ 
has subcritical growth.

On the other hand, Morse theory and local linking theorem are powerful tools 
in modern nonlinear analysis \cite{19, 18,20}, especially for the problems
 with resonance \cite{22,21}. However, there are no existed papers
 dealing with the existence of solutions for fractional Schr\"odinger 
equations by using Morse theory.

Motivated by the above facts, the goal of this paper is to consider the multiplicity
of nontrivial solutions for problem \eqref{e1.1}. Under some natural 
assumptions, by using Morse theory in combination with local linking 
arguments, the existence results of at least two nontrivial solutions are 
obtained. Next we prove that the problems have $k$ distinct pairs of solutions
 by using the Clark theorem. It is worthy stressing that we will use a more 
general assumption on $V(x)$, which extend some recent results from the literature.

Next we state our main results, using the following assumptions:
\begin{itemize}
\item[(V1)]  $V\in C(\mathbb{R}^{N},\mathbb{R})$ and 
$\beta:=\inf_{\mathbb{R}^{N}}V(x)>0$.

\item[(F1)] There exist constants $1<\gamma_{1}<\gamma_2<\dots<\gamma_{m}<2$
and positive functions $\xi_{1}(x) \in L^{\frac{2}{2-\gamma_{1}}}
(\mathbb{R}^{N},\mathbb{R})$, \dots, 
$\xi_{m}(x) \in L^{\frac{2}{2-\gamma_{m}}}(\mathbb{R}^{N},\mathbb{R})$ such that
$$
|f(x,u)|\leq \gamma_{1}\xi_{1}(x)|u|^{\gamma_{1}-1}+\dots
+\gamma_{m}\xi_{m}(x)|u|^{\gamma_{m}-1},\quad \forall (x,u)\in \mathbb{R}^{N} 
\times \mathbb{R}.
$$

\item[(F2)] There exist $c_{1}>0$, $0<c_2<\frac{1}{2S_2^2}$,
$1<\gamma<2$ and small constants $0<r<r_{0}$, such that
$$
c_2|u|^2>F(x,u)\geq c_{1} |u|^{\gamma},\quad  r\leq|u|\leq r_{0}
\quad \mbox{a.e.}\quad x\in\mathbb{R}^{N},
$$
where $S_2$ is the Sobolev constant from $H^{\alpha}(\mathbb{R}^{N})$ 
to $L^2(\mathbb{R}^{N})$; furthermore, in the sequel 
$F(x,u)=\int_{0}^{u}f(x,s)ds$.

\item[(F3)] $f(x,-u)=-f(x,u)$.
\end{itemize}

\begin{theorem} \label{thm1.1}
Assume that the potential $V(x)$ and the nonlinearity $f(x,u)$ satisfy 
{\rm (V1), (F1)--(F2)}.
 Then  problem \eqref{e1.1} has at least two nontrivial solutions..
\end{theorem}

\begin{theorem} \label{thm1.2}
Assume that {\rm (V1), (F1)--(F3)} are satisfied. 
Then problem \eqref{e1.1} has at least $k$ distinct pairs of solutions.
\end{theorem}

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

\section{Variational setting and preliminaries}

In this section, we collect some information to be used later.
We will denote either by  $\hat{u}$
or by $\mathcal{F}u$ the usual Fourier transform of $u$.

Sobolev spaces of fractional order are the convenient setting for our equations. 
A complete introduction to fractional Sobolev spaces can be found 
in \cite{Nezza}, we offer below a short review.
We recall that the fractional Sobolev space $W^{\alpha,p}(\mathbb{R}^{N})$ 
is defined for any $p\in[1,+\infty)$ and $\alpha\in(0,1)$ as
$$
W^{\alpha,p}(\mathbb{R}^{N})
=\big\{u\in L^{p}(\mathbb{R}^{N}): \int_{\mathbb{R}^{N}}
\frac{|u(x)-u(y)|^{p}}{|x-y|^{\alpha p+N}}\,dx\,dy<\infty\big\}.
$$
This space is endowed with the Gagliardo norm
$$
\|u\|_{W^{\alpha,p}}=\Big(\int_{\mathbb{R}^{N}}|u|^{p}\,dx
+\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{\alpha p+N}}\,dx\,dy
\Big)^{1/p}.
$$
When $p=2$, these spaces are also denoted by $H^{\alpha}(\mathbb{R}^{N})$.

If $p=$2, an equivalent definition of fractional Sobolev spaces is possible,
 based on Fourier analysis. Indeed, it turns out that
$$
H^{\alpha}(\mathbb{R}^{N})=\big\{u\in L^2(\mathbb{R}^{N}):
 \int_{\mathbb{R}^{N}}(1+|\xi|^{2\alpha})|\hat{u}|^2d\xi<\infty\big\},
$$
and the norm can be equivalently written by
$$
\|u\|_{H^{\alpha}}=\Big(\|u\|_2^2
+\int_{\mathbb{R}^{N}}|\xi|^{2\alpha}|\hat{u}|^2d\xi\Big)^{1/2}.
$$
Furthermore,  we know that $\|\cdot\|_{H^{\alpha}}$ is equivalent to the norm
$$
\|u\|_{H^{\alpha}}=\Big(\int_{\mathbb{R}^{N}}(|(-\Delta)^{\alpha/2}u|^2
+u^2)\,dx\Big)^{1/2}.
$$
In this article, in view of the potential $V(x)$, we consider its subspace
$$
E=\big\{u\in H^{\alpha}(\mathbb{R}^{N}): \int_{\mathbb{R}^{N}} V(x)u^2\,dx
<\infty\big\}.
$$
Then, by \cite{Secchi}, $E$ is a Hilbert space with the inner product
$$
(u,v)_{E}=\int_{\mathbb{R}^{N}}(|\xi|^{2\alpha}\hat{u}(\xi)\hat{v}(\xi)
+\hat{u}(\xi)\hat{v}(\xi))d\xi+\int_{\mathbb{R}^{N}}V(x)u(x)v(x)\,dx, \quad
 \forall u,v \in E,
$$
 and the norm
$$
\|u\|_{E}=\Big(\int_{\mathbb{R}^{N}}(|\xi|^{2\alpha}\hat{u}^2
+\hat{u}^2)d\xi+ \int_{\mathbb{R}^{N}} V(x)u^2\,dx\Big)^{1/2}.
$$
Furthermore,  we know that $\|\cdot\|_{E}$ is equivalent to the norm
$$
\|u\|=\Big(\int_{\mathbb{R}^{N}}(|(-\Delta)^{\alpha/2}u|^2+V(x)u^2)\,dx
\Big)^{1/2}.
$$
The corresponding inner product is
$$
(u,v)=\int_{\mathbb{R}^{N}}((-\Delta)
^{\alpha/2}u(x)(-\Delta)^{\alpha/2}v(x)+V(x)u(x)v(x))\,dx.
$$
Throughout out this paper, we  use the norm $\|\cdot\|$ in $E$.

As usual, for $1\leq p< +\infty$, we let
\begin{gather*}
\|u\|_{p}=\Big(\int_{\mathbb{R}^{N}}|u(x)|^{p}\,dx\Big)^{1/p},\quad
u\in L^{p}(\mathbb{R}^{N}), \\
\|u\|_{\infty}=\mbox{ess sup}_{x\in \mathbb{R}^{N}} |u(x)|, \quad 
u\in L^{\infty}(\mathbb{R}^{N}).
\end{gather*}
To prove our results, the following compactness result is necessary.

\begin{lemma}[\cite{Lions}] \label{lem2.1}
$E$ is continuously embedded into $L^{p}(\mathbb{R}^{N})$ for
 $2\leq p \leq 2_{\alpha}^{\ast}$ and compactly embedded into 
$L_{loc}^{p}(\mathbb{R}^{N})$ for $2\leq p <2_{\alpha}^{\ast}$ with
 $2_{\alpha}^{\ast}=\frac{2N}{N-2\alpha}$.
\end{lemma}

It follows directly from the Lemma \ref{lem2.1} that there are constants
 $S_{p}>0$ such that
$$
\|u\|_{p}\leq S_{p}\|u\|, \quad \forall u\in E, \; p\in [2,2_{\alpha}^{\ast}].
$$

\begin{lemma} \label{lem2.2}
Assume that {\rm (V1), (F1)} hold. Then the functional
 $\Phi: E\to \mathbb{R}$ defined by
\begin{equation}
\Phi(u)=\frac{1}{2}\|u\|^2-\int_{\mathbb{R}^{N}}F(x,u(x))\,dx\label{e2.1}
\end{equation}
is well defined and of class $C^{1}(E,\mathbb{R})$ and
\begin{equation}
(\Phi'(u),v)=(u,v)-\int_{\mathbb{R}^{N}}f(x,u(x))v(x)\,dx.\label{e2.2}
\end{equation}
Furthermore, the critical points of $\Phi$ in $E$ are solutions of
 problem \eqref{e1.1}.
\end{lemma}

\begin{proof} 
 From (F1), one has
\begin{equation}
|F(x,u)|\leq \xi_{1}(x)|u(x)|^{\gamma_{1}}+\dots
+\xi_{m}(x)|u(x)|^{\gamma_{m}}, \quad \forall (x,u)\in\mathbb{R}^{N}\times\mathbb{R}.
\label{e2.3}
\end{equation}
For any $u\in E$,  from (V1), \eqref{e2.3} and the
 H\"older inequality, it follows that
\begin{align*}
\int_{\mathbb{R}^{N}}|F(x,u)|\,dx
&\leq\int_{\mathbb{R}^{N}}[\xi_{1}(x)|u(x)|^{\gamma_{1}}
 +\dots+\xi_{m}(x)|u(x)|^{\gamma_{m}}]\,dx\\
&\leq\sum_{i=1}^{m}\beta^{-\gamma_i/2}
\Big(\int_{\mathbb{R}^{N}}|\xi_{i}(x)|^{\frac{2}{2-\gamma_{i}}}\,dx\Big)
 ^{\frac{2-\gamma_{i}}{2}}\Big(\int_{\mathbb{R}^{N}}V(x)|u(x)|^2\,dx
 \Big)^{\gamma_i/2}\\
&\leq\sum_{i=1}^{m}\beta^{-\gamma_i/2}\|\xi_{i}\|
 _{\frac{2}{2-\gamma_{i}}}\|u\|^{\gamma_{i}},
\end{align*}
and so $\Phi$ defined by \eqref{e2.1} is well defined on $E$.

Next, we prove that \eqref{e2.2} holds. 
For any function $\theta : \mathbb{R}\to(0,1)$, by (F1) and the H\"older 
inequality, we have
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}^{N}}\max_{t\in[0,1]}|f(x,u(x)+t\theta(x)v(x))v(x)|\,dx\\
& \leq\int_{\mathbb{R}^{N}}\max_{t\in[0,1]}|f(x,u(x)+t\theta(x)v(x))||v(x)|\,dx\\
& \leq\sum_{i=1}^{m}\gamma_{i}\int_{\mathbb{R}^{N}}\xi_{i}(x)(|u(x)|
 +|v(x)|)^{\gamma_{i}-1}|v(x)|\,dx\\
&\leq\sum_{i=1}^{m}\gamma_{i}\int_{\mathbb{R}^{N}}\xi_{i}(x)
 (|u(x)|^{\gamma_{i}-1}+|v(x)|^{\gamma_{i}-1})|v(x)|\,dx\\
&\leq\sum_{i=1}^{m}\gamma_{i}\beta^{-\gamma_i/2}
 \Big(\int_{\mathbb{R}^{N}}|\xi_{i}(x)|^{\frac{2}{2-\gamma_{i}}}\,dx
 \Big)^{\frac{2-\gamma_{i}}{2}}\Big(\int_{\mathbb{R}^{N}}V(x)|u(x)|^2\,dx
 \Big)^{\frac{\gamma_{i}-1}{2}}\\
&\quad \times\Big(\int_{\mathbb{R}^{N}}V(x)|v(x)|^2\,dx\Big)^{1/2}\\
&\quad +\sum_{i=1}^{m}\gamma_{i}\beta^{-\gamma_i/2}
 \Big(\int_{\mathbb{R}^{N}}|\xi_{i}(x)|^{\frac{2}{2-\gamma_{i}}}\,dx
 \Big)^{\frac{2-\gamma_{i}}{2}}
 \Big(\int_{\mathbb{R}^{N}}V(x)|u(x)|^2\,dx\Big)^{\gamma_i/2}\\
&\leq\sum_{i=1}^{m}\gamma_{i}\beta^{-\gamma_i/2}\|\xi_{i}
 \|_{\frac{2}{2-\gamma_{i}}}(\|u\|^{\gamma_{i}-1}+\|v\|^{\gamma_{i}-1})\|v\|
<+\infty.
\end{aligned} \label{e2.4}
\end{equation}
Then by the above inequality, \eqref{e2.1} and the Lebesgue's Dominated
Convergence Theorem, we have
\begin{align*}
(\Phi'(u),v)
&=\lim_{t\to0^{+}}\frac{\Phi(u+tv)-\Phi(u)}{t}\\
&=\lim_{t\to0^{+}}\frac{1}{t}\Big\{\frac{\|u+tv\|^2  -  \|u\|^2}{2}
-\int_{\mathbb{R}^{N}}[F(x,u(x)  +  tv(x))  -  F(x,u(x))]\,dx\Big\}\\
&=\lim_{t\to0^{+}}\Big[(u,v)+\frac{t\|v\|^2}{2}
 -\int_{\mathbb{R}^{N}}f(x, u(x)+t\theta(x)v(x))v(x)\,dx\Big]\\
&=(u,v)-\int_{\mathbb{R}^{N}}f(x,u(x))v(x)\,dx.
\end{align*}
This shows that \eqref{e2.2} holds. Furthermore, by a standard argument,
it is easy to show that the critical points of $\Phi$ in $E$ are solutions
of problem \eqref{e1.1} (see\cite{Willem}).

Let us prove that $\Phi'$ is continuous. Let $u_{k}\to u$ in $E$,
 then $u_{k}\to u$ in $L^2(\mathbb{R}^{N})$, and so
\begin{equation}
\lim_{k\to\infty}u_{k}(x)=u(x), \quad \mbox{a.e. }x\in\mathbb{R}^{N}.\label{e2.5}
\end{equation}
We claim that
\begin{equation}
\lim_{k\to\infty}\int_{\mathbb{R}^{N}}|f(x,u_{k}(x))-f(x,u(x))|^2\,dx=0.\label{e2.6}
\end{equation}
Indeed, if it is not true, then there exists a constant $\varepsilon>0$ and
a subsequence $u_{k_{i}}$ such that
\begin{equation}
\int_{\mathbb{R}^{N}}|f(x,u_{k_{i}}(x))-f(x,u(x))|^2\,dx\geq\varepsilon,
\quad \forall i\in\mathbb{N}.\label{e2.7}
\end{equation}
Since $u_{k}\to u$ in $L^2(\mathbb{R}^{N})$, passing to a subsequence if
 necessary, it can be assumed that
$\sum_{i=1}^{\infty}\|u_{k_{i}}-u\|^2_2<+\infty$. Set
$\omega(x)=\left[\sum_{i=1}^{\infty}|u_{k_{i}}(x)-u(x)|^2\right]^{1/2}$,
$x\in\mathbb{R}^{N}$. Then $\omega \in L^2(\mathbb{R}^{N})$. Note that
\begin{equation}
\begin{aligned}
&|f(x,u_{k_{i}}(x))-f(x,u(x))|^2\\
&\leq 2|f(x,u_{k_{i}}(x))|^2+2|f(x,u(x))|^2\\
&\quad \leq 4\gamma_{1}^2|\xi_{1}(x)|^2\big[|u_{k_{i}}(x)|^{2
 (\gamma_{1}-1)}+|u(x)|^{2(\gamma_{1}-1)}\big]\\
&\quad +\dots+4\gamma_{m}^2|\xi_{m}(x)|^2\big[|u_{k_{i}}(x)|^{2(\gamma_{m}-1)}
 +|u(x)|^{2(\gamma_{m}-1)}\big]\\
&\leq \sum_{j=1}^{m}(4^{\gamma_{j}}+4)\gamma_{j}^2|\xi_{j}(x)|^2
\big[|u_{k_{i}}(x)-u(x)|^{2(\gamma_{j}-1)}+|u(x)|^{2(\gamma_{j}-1)}\big]\\
&\leq \sum_{j=1}^{m}(4^{\gamma_{j}}+4)\gamma_{j}^2|\xi_{j}(x)|^2
\big[|\omega(x)|^{2(\gamma_{j}-1)}+|u(x)|^{2(\gamma_{j}-1)}\big]\\
&:=g(x), \quad \forall i\in \mathbb{N},x\in\mathbb{R}^{N}
\end{aligned}\label{e2.8}
\end{equation}
and
\begin{equation}
\begin{aligned}
\int_{\mathbb{R}^{N}}g(x)\,dx
&=\sum_{j=1}^{m}(4^{\gamma_{j}}+4)\gamma_{j}^2\int_{\mathbb{R}^{N}}
|\xi_{j}(x)|^2\big[|\omega(x)|^{2(\gamma_{j}-1)}+|u(x)|^{2(\gamma_{j}-1)}\big]\,dx\\
&\leq \sum_{j=1}^{m}(4^{\gamma_{j}}+4)\gamma_{j}^2\|\xi_{j}
\|_{\frac{2}{2-\gamma_{j}}}^2\Big(\|\omega\|_2^{2(\gamma_{j}-1)}
+\|u\|_2^{2(\gamma_{j}-1)}\Big)
<+\infty.
\end{aligned}\label{e2.9}
\end{equation}
Then by \eqref{e2.5}, \eqref{e2.8}, \eqref{e2.9} and the Lebesgue's Dominated
Convergence Theorem, we have
$$
\lim_{i\to\infty}\int_{\mathbb{R}^{N}}|f(x,u_{k_{i}}(x))-f(x,u(x))|^2\,dx=0,
$$
which contradicts \eqref{e2.7}. Hence \eqref{e2.6} holds.
From \eqref{e2.2}, \eqref{e2.6} and the H\"older inequality, we have
\begin{align*}
&|(\Phi'(u_{k})-\Phi'(u),v)|\\
&=\Big|(u_{k}-u,v)-\int_{\mathbb{R}^{N}}[f(x,u_{k}(x))-f(x,u(x))]v(x)\,dx\Big|\\
&\leq\|u_{k}-u\|\|v\|+\int_{\mathbb{R}^{N}}|f(x,u_{k}(x))-f(x,u(x))||v(x)|\,dx\\
&\leq  \|u_{k}-u\|\|v\|  +  \beta^{\frac{-1}{2}}
\Big(\int_{\mathbb{R}^{N}}|f(x,u_{k}(x)) - f(x,u(x))|^2\,dx\Big)^{1/2}\|v\|
=o(1),
\end{align*}
as $k\to+\infty$, which implies the continuity of $\Phi'$.
The proof is complete.
\end{proof}

We will use Morse theory in combination with local linking arguments to obtain 
the critical points of $\Phi$. Now, it is necessary to recall the following 
definitions and results.

 \begin{definition} \label{def2.1}\rm
 Let $E$ be a real reflexive Banach space. We say that $\Phi$ satisfies 
the (PS)-condition, i.e. every sequence $\{u_{n}\}\subset E$ satisfying 
$\Phi(u_{n})$ bounded and $\lim_{n\to\infty}\Phi'(u_{n})=0$ contains 
a convergent subsequence.
\end{definition}

Let $E$ be a real Banach space and $\Phi\in C^{1}(E,\mathbb{R})$. 
$K=\{u\in E:\Phi'(u)=0\}$, then the $q$th critical group of $\Phi$ at an 
isolated critical point $u\in K$ with $\Phi(u)=c$ is defined by
$$
C_{q}(\Phi,u):=H_{q}(\Phi^{c}\cap U,\Phi^{c}\cap U\setminus \{u\}), 
\quad q\in \mathbb{N}:=\{0,1,2,\dots\},
$$
where $\Phi^{c}=\{u\in E: \Phi (u)\leq c\}$, $U$ is a neighborhood of $u$, 
containing the unique critical point, $H_{*}$ is the singular relative 
homology with coefficient in an Abelian group $G$.

We say that $u\in E$ is a homological nontrivial critical point of $\Phi$ 
if at least one of its critical groups is nontrivial.
Now, we present the following propositions which will be used later.

\begin{proposition}[{\cite[Proposition 2.1]{26}}] \label{prop2.1}
Assume that $\Phi$ has a critical point $u=0$ with $\Phi(0)=0$. Suppose that 
$\Phi$ has a local linking at 0 with respect to $E=V\oplus W$, 
$k=\operatorname{dim}V<\infty$; that is, there exists $\rho >0$ small such that
\begin{gather*} %\label{1.1}
\Phi(u)\leq 0,\quad u\in V,\; \|u\|\leq\rho;\\
\Phi(u)> 0,\quad u\in W, \;  0<\|u\|\leq\rho.
 \end{gather*}
Then $C_{k}(\Phi,0)\ncong0$, hence 0 is a homological nontrivial critical
 point of $\Phi$.
\end{proposition}

\begin{proposition}[{\cite[Theorem 2.1]{26}}] \label{prop2.2}
Let $E$ be a real Banach space and let $\Phi\in C^{1}(E,\mathbb{R})$ satisfy the 
(PS)-condition and is bounded from below. If  $\Phi$ has a critical point that 
is homological nontrivial and is not a minimizer of  $\Phi$, then $\Phi$ 
has at least three critical points.
\end{proposition}

\begin{proposition}[{\cite[Theorem 9.1]{P.H.}}] \label{prop2.3}
Let $E$ be a real Banach space, $\Phi\in C^{1}(E,\mathbb{R})$ with $\Phi$ even,
 bounded from below, and satisfying (PS)-condition. Suppose $\Phi(0)=0$, 
there is a set $K\subset E$ such that $K$ is homeomorphic to $S^{j-1}$ 
by an odd map, and $\sup_{K}\Phi<0$. Then $\Phi$ possesses at least 
$j$ distinct pairs of critical points.
\end{proposition}

 \section{Proof of main results}

In this section, we  prove Theorems \ref{thm1.1} and \ref{thm1.2}. To this end  
we need the following lemmas.
 
  \begin{lemma} \label{lem3.1} 
Suppose that $\Phi$ satisfies {\rm (V1)} and {\rm (F1)}, then $\Phi$ satisfies 
the (PS)-condition.
\end{lemma}

\begin{proof}
 We first prove that $\Phi$ is coercive. By \eqref{e2.1}, \eqref{e2.3} and the
 H\"older inequality, we have
\begin{equation}
\begin{aligned}
\Phi(u)&=\frac{1}{2}\|u\|^2-\int_{\mathbb{R}^{N}}F(x,u(x))\,dx\\
&\geq \frac{1}{2}\|u\|^2-\sum_{i=1}^{m}
 \int_{\mathbb{R}^{N}}\xi_{i}(x)|u(x)|^{\gamma_{i}}\,dx\\
&\geq \frac{1}{2}\|u\|^2-\sum_{i=1}^{m}\beta^{-\gamma_i/2}
\Big(\int_{\mathbb{R}^{N}}|\xi_{i}(x)|^{\frac{2}{2-\gamma_{i}}}\,dx
 \Big)^{\frac{2-\gamma_{i}}{2}}
 \Big(\int_{\mathbb{R}^{N}}V(x)|u(x)|^2\,dx\Big)^{\gamma_i/2}\\
&\geq \frac{1}{2}\|u\|^2-\sum_{i=1}^{m}\beta^{-\gamma_i/2}
 \|\xi_{i}\|_{\frac{2}{2-\gamma_{i}}}\|u\|^{\gamma_{i}}.
\end{aligned}\label{e3.1}
\end{equation}
Since $1<\gamma_{1}<\dots<\gamma_{m}<2$, \eqref{e3.1} implies that
$\Phi(u)\to+\infty$ as $\|u\|\to+\infty$.

Next, we prove that $\Phi$ satisfies the (PS)-condition. Assume that
 $\{u_{k}\}\subset E$ is a sequence such that $\{\Phi(u_{k})\}$ is 
bounded and $\Phi'(u_{k})\to 0$ as $k\to+\infty$. Then by \eqref{e3.1}, there
exists a constant $M>0$ such that
\begin{equation}
 \|u_k\|\leq M,\quad\forall k\in \mathbb{N}. \label{e3.2}
\end{equation}
Going if necessary to a subsequence we can assume that $u_k\rightharpoonup u_0$
in $E$. For any given number $\varepsilon>0$, by (F1), we can choose
$R_{\varepsilon}>0$ such that
\begin{equation}
\Big(\int_{|x|>R_{\varepsilon}}|\xi_{i}(x)|^{\frac{2}{2-\gamma_{i}}}
\Big)^{\frac{2-\gamma_{i}}{2}}<\varepsilon, \quad i=1,2,\dots,n.\label{e3.3}
\end{equation}
Since the embedding of $E\hookrightarrow L_{loc}^2(\mathbb{R}^{N})$
is compact, then
$$
u_{k}\to u_{0}, \quad \mbox{in}\  L_{loc}^2(\mathbb{R}^{N}),
$$
and hence,
\begin{equation}
\lim_{k\to\infty}\int_{|x|\leq R_{\varepsilon}}|u_{k}(x)-u_{0}(x)|^2\,dx=0.
\label{e3.4}
\end{equation}
By \eqref{e3.4}, there exists $k_{0}\in \mathbb{N}$ such that
\begin{equation}
\int_{|x|\leq R_{\varepsilon}}|u_{k}(x)-u_{0}(x)|^2\,dx<\varepsilon^2,
\quad \mbox{for }k\geq k_{0}. \label{e3.5}
\end{equation}
Hence, by (F1), \eqref{e3.2}, \eqref{e3.5} and the H\"older inequality,
for any $k\geq k_{0}$ we have
\begin{equation}
\begin{aligned}
 &\int_{|x|\leq R_{\varepsilon}}|f(x,u_{k}(x))-f(x,u_{0}(x))||u_{k}(x)-u_{0}(x)|\,dx\\
&\leq\Big(\int_{|x|\leq R_{\varepsilon}}|f(x,u_{k}(x))-f(x,u_{0}(x))|^2\,dx\Big)^{1/2}
 \Big(\int_{|x|\leq R_{\varepsilon}}|u_{k}(x)-u_{0}(x)|^2\,dx\Big)^{1/2}\\
&\leq\Big[\int_{|x|\leq R_{\varepsilon}}2(|f(x,u_{k}(x))|^2
 +|f(x,u_{0}(x))|^2)\,dx\Big]^{1/2}\varepsilon\\
&\leq2\Big[\sum_{i=1}^{m}\gamma_{i}^2\int_{|x|\leq R_{\varepsilon}}
 |\xi_{i}(x)|^2(|u_{k}(x)|^{2(\gamma_{i}-1)}+|u_{0}(x)|^{2(\gamma_{i}-1)})\,dx
 \Big]^{1/2}\varepsilon\\
&\leq2\Big[\sum_{i=1}^{m}\gamma_{i}^2\|\xi_{i}\|^2_{\frac{2}{2-\gamma_{i}}}
 (\|u_{k}\|_2^{2(\gamma_{i}-1)}+\|u_{0}\|_2^{2(\gamma_{i}-1)})\Big]^{1/2}\varepsilon\\
&\leq 2\Big[\sum_{i=1}^{m}\gamma_{i}^2\|\xi_{i}\|^2_{\frac{2}{2-\gamma_{i}}}
 (M^{2(\gamma_{i}-1)}+\|u_{0}\|_2^{2(\gamma_{i}-1)})\Big]^{1/2}\varepsilon.
\end{aligned}\label{e3.6}
\end{equation}
On the other hand, for $k\in\mathbb{N}$, it follows from (F1), \eqref{e3.2},
\eqref{e3.3} and the H\"older inequality that
\begin{equation}
\begin{aligned}
 &\int_{|x|> R_{\varepsilon}}|f(x,u_{k}(x))-f(x,u_{0}(x))||u_{k}(x)-u_{0}(x)|\,dx\\
&\leq \sum_{i=1}^{m}\gamma_{i}\int_{|x|> R_{\varepsilon}}|\xi_{i}(x)|
 (|u_{k}(x)|^{\gamma_{i}-1}+|u_{0}(x)|^{\gamma_{i}-1})(|u_{k}(x)|+|u_{0}(x)|)\,dx\\
&\leq2\sum_{i=1}^{m}\gamma_{i}\int_{|x|> R_{\varepsilon}}|\xi_{i}(x)|
 (|u_{k}(x)|^{\gamma_{i}}+|u_{0}(x)|^{\gamma_{i}})\,dx\\
&\leq2\sum_{i=1}^{m}\gamma_{i}\Big(\int_{|x|> R_{\varepsilon}}|\xi_{i}(x)
 |^{\frac{2}{2-\gamma_{i}}}\,dx\Big)^{\frac{2-\gamma_{i}}{2}}
 (\|u_{k}\|_2^{\gamma_{i}}+\|u_{0}\|_2^{\gamma_{i}})\\
&\leq2\sum_{i=1}^{m}\gamma_{i}\Big(\int_{|x|> R_{\varepsilon}}|\xi_{i}(x)
 |^{\frac{2}{2-\gamma_{i}}}\,dx\Big)^{\frac{2-\gamma_{i}}{2}}(M^{\gamma_{i}}
 +\|u_{0}\|_2^{\gamma_{i}})\\
&\leq2\sum_{i=1}^{m}\gamma_{i}(M^{\gamma_{i}}+\|u_{0}\|_2^{\gamma_{i}})\varepsilon.
\end{aligned} \label{e3.7}
\end{equation}
Since $\varepsilon$ is arbitrary, combining \eqref{e3.6} with \eqref{e3.7},
one has
\begin{equation}
\aligned\int_{\mathbb{R}^{N}}[f(x,u_{k}(x))-f(x,u_{0}(x))][u_{k}(x)-u_{0}(x)]\,dx\to0
\endaligned\label{e3.8}
\end{equation}
as $k\to\infty$. It follows from \eqref{e2.2} that
\begin{equation}
\begin{aligned}
&(\Phi'(u_{k})-\Phi'(u_{0}),u_{k}-u_{0})\\
&=\|u_{k}-u_{0}\|^2
-\int_{\mathbb{R}^{N}}[f(x,u_{k}(x))-f(x,u_{0}(x))][u_{k}(x)-u_{0}(x)]\,dx.
\end{aligned} \label{e3.9}
\end{equation}
In view of the definition of weak convergence, we have
\begin{equation}
(\Phi'(u_{k})-\Phi'(u_{0}),u_{k}-u_{0})=0. \label{e3.10}
\end{equation}
It follows from \eqref{e3.8}-\eqref{e3.10} that
$$
u_{k}\to u_{0}   \quad\text{in }E.
$$
Hence, $\Phi$ satisfies the (PS)-condition.
\end{proof}

We choose an orthogonal basis $\{e_{j}\}$ of $E$ and define 
$X_{j}:=\operatorname{span}\{e_{j}\},j=1,2,\dots$, 
$Y_{k}:=\oplus_{j=1}^{k} X_{j}$, 
$Z_{k}=\overline{\oplus_{j=k+1}^{\infty} X_{j}}$, then $E=Y_{k}\oplus Z_{k}$.

\begin{lemma} \label{lem3.2} 
Suppose that the conditions of Theorem \ref{thm1.1} are satisfied, then there 
exists $k_{0}\in \mathbb{N}$ such that $C_{k_{0}}(\Phi, 0)\ncong0$.
\end{lemma}

\begin{proof}
 It follows from (F1) that the zero function is a critical point of $\Phi$. 
So we only need to prove that $\Phi$ has a local linking at 0 with respect 
to $E=Y_{k}\oplus Z_{k}$.
\smallskip

\noindent\textbf{Step 1:}
 Take $u\in Y_{k}$, since $Y_{k}$ is finite dimensional, we have that for given 
$r_{0}$, there exists $0<\rho<1$ small such that
$$
u\in Y_{k}, \; \|u\|\leq\rho\Rightarrow |u|<r_{0}, \quad x\in \mathbb{R}^{N}
$$
For $0<r<r_{0}$, let $\Omega_{1}=\{x\in\mathbb{R}^{N}:|u(x)|<r\}$, 
$\Omega_2=\{x\in\mathbb{R}^{N}:r\leq|u(x)|\leq r_{0}\}$, 
$\Omega_3=\{x\in\mathbb{R}^{N}:|u(x)|>r_{0}\}$, then 
$\mathbb{R}^{N}=\bigcup_{i=1}^{3}\Omega_{i}$. For the sake of simplicity, 
let $G(x,u)=F(x,u)-c_{1}|u|^{\gamma}$. Therefore, form (F2) it follows that
\begin{align*} 
 \Phi(u)&=\frac{1}{2}\|u\|^2-\int_{\mathbb{R}^{N}}c_{1}|u|^{\gamma}\,dx
-\Big(\int_{\Omega_{1}}+\int_{\Omega_2}+\int_{\Omega_3}\Big)G(x,u)\,dx\\
&\leq\frac{1}{2}\|u\|^2-\int_{\mathbb{R}^{N}}c_{1}|u|^{\gamma}\,dx
-\int_{\Omega_{1}}G(x,u)\,dx.
\end{align*}

Note that the norms on $Y_{k}$ are equivalent to each other,
 $\|u\|_{\gamma}$ is equivalent to $\|u\|$ and 
$\int_{\Omega_{1}}G(x,u)\,dx\to0$ as $r\to0$.
Since $0<\gamma<2$, then $\Phi(u)\leq0$, for all $u\in Y_{k}$ with $\|u\|\leq\rho$.
\smallskip

\noindent\textbf{Step 2:}
 Take $u\in Z_{k}$, since the embedding $E\hookrightarrow L^{p}$ is continuous, 
we have that for given $r_{0}$, there exists $0<\rho<1$ small such that
$$
u\in Z_{k}, \; \|u\|\leq\rho\Rightarrow |u|<r_{0}, \quad x\in \mathbb{R}^{N}.
$$
Therefore, it follows from  (F2) that
$$
 \Phi(u) \geq\frac{1}{2}\|u\|^2-\int_{\mathbb{R}^{N}}c_2|u|^2\,dx\\
>\frac{1}{2}\|u\|^2-\frac{1}{2}\|u\|^2=0.
$$
 Therefore, by Proposition \ref{prop2.1}, the proof is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1.1}]
By Lemma \ref{lem3.1}, $\Phi$ satisfies the (PS)-condition and is bounded from below. 
By Lemma \ref{lem3.2} and Proposition \ref{prop2.1}, the trivial solution $u=0$ is homological 
nontrivial and is not a minimizer. Then Theorem \ref{thm1.1} follows immediately 
from Proposition \ref{prop2.2}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]  
By (F3), we can easily check that the functional $\Phi$ is even. 
Lemma \ref{lem3.1} shows that $\Phi$ satisfies the (PS)-condition and is bounded 
from below. For $\rho>0$, let $K=S_{\rho}=\{u\in Y_{k}:\|u\|=\rho\}$. 
Thus, just as shown in the proof of Lemma \ref{lem3.2}, if $\rho>0$ is small enough,
 we have that
$$
\sup_{K}\Phi(u)\leq 0.
$$
By the definition of $Y_{k}$, we have $\dim Y_{k}=k$, then by
 Proposition \ref{prop2.3}, 
we have that $\Phi$ has at least $k$ distinct pairs of critical points. 
Therefore, problem \eqref{e1.1} has at least $k$ distinct pairs of solutions.
\end{proof}

\subsection*{Acknowledgments}
This work is partially supported by Natural Science Foundation of China 11271372
and by Hunan Provincial Natural Science Foundation of China 12JJ2004.


 \begin{thebibliography}{00}

\bibitem{T} T. Bartsch, A. Pankov, Z. Q. Wang;
\newblock{Nonlinear Schr\"odinger equations with steep potential well,}
\newblock{\em Commun. Contemp. Math.} \textbf{3} (2001) 549-569.

\bibitem{X}  X. Chang;
 \newblock{Ground state solutions of asymptotically linear fractional 
Schr\"odinger equation,}
 \newblock{\em J. Math. Phys.} \textbf{54 }(2013) 061504.

\bibitem{19}  S. Chen, C. Wang;
 \newblock{Existence of multiple nontrivial solutions for a 
Schr\"odinger-Poisson system,}
\newblock {\em J. Math. Anal. Appl.} \textbf{411} (2014) 787-793.

\bibitem{Dipierro} S. Dipierro, G. Palatucci, E. Valdinoci;
\newblock{Existence and symmetry results for a Schr\"odinger type problem
involving the fractional Laplacian,}
\newblock{\em Le matematiche} \textbf{68} (2013) 201-216.

\bibitem{Nezza} E. Di Nezza, G. Palatucci, E. Valdinoci;
\newblock{Hitchhiker¡¯s guide to the fractional Sobolev spaces,}
\newblock{\em Bull. Sci. Math.} \textbf{136(5)} (2012) 521-573.

\bibitem{Felmer} P. Felmer, A. Quaas, J. G. Tan;
\newblock{Positive solutions of nonlinear Schr\"odinger equation with the 
fractional Laplacian,}
\newblock {\em Proc. R. Soc. Edinburgh, Sect. A: Math.} \textbf{142} (2012) 1237-1262.

\bibitem{B1} G. Molica Bisci;
 \newblock{Sequences of weak solutions for fractional equations,}
\newblock {\em Math. Res. Lett.} \textbf{21(2)} (2014) 241-253.

\bibitem{B2} G. Molica Bisci, B. A. Pansera;
\newblock{Three weak solutions for nonlocal fractional equations,}
\newblock {\em Adv. Nonlinear Stud.} \textbf{14(3)} (2014) 619-629.

\bibitem{B3} G. Molica Bisci, D. Repov\v{s};
 \newblock{Higher nonlocal problems with bounded potential,}
\newblock {\em J. Math. Anal. Appl.} \textbf{420(1)} (2014) 167-176.

\bibitem{18}  M. Jiang,  M. Sun;
 \newblock{Some qualitative results of the critical groups for the p-Laplacian 
equations,} 
\newblock {\em Nonlinear Analysis: TMA} \textbf{75} (2012) 1778-1786.

\bibitem{22} K. Li, S. Wang, Y. Zhao;
\newblock{Multiple periodic solutions for asymptotically linear Duffing 
equations with resonance (II),}
\newblock{\em J. Math. Anal. Appl.} \textbf{397} (2013) 156-160.

\bibitem{Li} Y. Q. Li, Z. Q. Wang, J. Zeng;
 \newblock{Ground states of nonlinear Schr\"odinger equations with potentials,}
 \newblock{ \em Ann. Inst. Henri Poincar\'{e}, Anal. Non Lineair\'{e}} 
\textbf{23} (2006) 829-837.

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

\bibitem{PL} P. L. Lions;
\newblock{The concentration-compactness principle in the calculus of variations. 
The locally compact case Part I,} 
 \newblock{\em Ann. Inst. Henri Poincar\'{e}, Anal. Non Lineair\'{e}} \textbf{1} (1984) 109-145.

\bibitem{26} J. Q. Liu, J. B. Su;
\newblock{Remarks on multiple nontrivial solutions for quasi-linear resonant 
problems,} \newblock{\em J. Math. Anal. Appl.} \textbf{258} (2001) 209-222.

\bibitem{Willem} M. Millem;
\newblock{Minimax Theorems,}
 \newblock{\em Birkh$\ddot{a}$user, Berlin,} 1996.

\bibitem{P.H.} P. H. Rabinowitz;
\newblock{Minimax Methods in Critical Point Theory with Applicaions to 
Differential Equations,} \newblock{\em in: CBMS Reg. Conf. Ser. in Math., Vol.
65, Amer. Math. Soc., Providence, RI,} 1986.

 \bibitem{Rabin}  P. H. Rabinowitz;
\newblock{On a class of nonlinear Schr\"odinger equations,}
 \newblock{\em Z. Angew. Math. Phys.} \textbf{43} (1992) 270-291.

  \bibitem{Ricceri}   B. Ricceri;
 \newblock{A multiplicity result for nonlocal problems involving nonlinearities 
with bounded primitive,}
 \newblock{\em Studia Univ. Babes-Bolyai Math.} \textbf{55} (2010) 107-114.

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

\bibitem{S} S. Secchi;
 \newblock{On fractional Schr\"odinger equations in $\mathbb{R}^{N}$ without 
the Ambrosetti-Rabinowitz condition,} (2014) arXiv:1210.0755v2.

\bibitem{Shang} X. Shang, J. Zhang;
\newblock{Ground states for fractional Schr\"odinger equations with critical growth,}
\newblock{\em Nonlinearity} \textbf{27} (2014) 187-207.

\bibitem{21} J. B. Su;
\newblock{Semilinear elliptic boundary value problems with double resonance 
between two consecutive eigenvalues,}
\newblock{\em Nonlinear Analysis} \textbf{48} (2002) 881-895.

 \bibitem{20} M. Sun;
 \newblock{Multiplicity of solutions for a class of the quasilinear elliptic 
equations at resonance,}
\newblock {\em J. Math. Anal. Appl.} \textbf{386} (2012) 661-668.

\bibitem{A} A. Szulkin, T. Weth;
 \newblock{ Ground state solutions for some indefinite variational problems,}
  \newblock{\em J. Funct. Anal.}\textbf{ 257 }(2009) 3802-3822.

\bibitem{Tang2} X. H. Tang;
\newblock{Infinitely many solutins for semilinear Schr\"odinger equation with
 sign-changing potential and nonlinearity,} \newblock{\em J. Math. Anal. Appl.} 
\textbf{401} (2013) 407-415.

\bibitem{Tang} X. H. Tang;
\newblock{New conditions on nonlinearity for a periodic Schr\"odinger equation
 having zero as spectrum,}
\newblock{\em J. Math. Anal. Appl.} \textbf{413} (2014) 392-410.

\bibitem{Tang1} X. H. Tang;
\newblock{New super-quadratic conditions on ground state solutions for  
superlinear Schr\"odinger equation,}
\newblock{\em Adv. Nonlinear Studies} \textbf{14} (2014) 349-361.

\bibitem{Wei} D. Wei, J. Xu, Z. Wei;
\newblock{Infinitely many weak solutions for a fractional Schr\"odinger equation,}
\newblock{\em Boundary Value Problems} (2014) 2014:159.

\bibitem{Zou} W. M. Zou;                                                                                                              Zou,
\newblock{Variant fountain theorems and their applications,}
\newblock{\em Manuscripta Math.} \textbf{104} (2001) 343-358.

\end{thebibliography}

\end{document}