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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 38, pp. 1--9.\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-2017/38\hfil Multiplicity of ground state solutions]
{Multiplicity of ground state solutions for discrete
nonlinear Schr\"odinger equations with unbounded potentials}

\author[X. Liu, T. Zhou, H. Shi \hfil EJDE-2017/38\hfilneg]
{Xia Liu, Tao Zhou, Haiping Shi}

\address{Xia Liu \newline
Oriental Science and Technology College,
Hunan Agricultural University,
Changsha 410128, China. \newline
Science College, Hunan Agricultural University,
 Changsha 410128, China}
\email{xia991002@163.com}

\address{Tao Zhou \newline
School of Business Administration,
South China University of Technology,
Guangzhou 510640, China}
\email{zhoutaoscut@hotmail.com}

\address{Haiping Shi \newline
Modern Business and Management Department,
Guangdong Construction Polytechnic,
Guangzhou 510440, China}
\email{shp7971@163.com}

\dedicatory{Communicated by Paul H. Rabinowitz}

\thanks{Submitted September 8, 2016. Published February 2, 2017.}
\subjclass[2010]{39A12, 39A70, 35C08}
\keywords{Ground state solutions; critical point theory;
\hfill\break\indent discrete nonlinear Schr\"odinger equation}

\begin{abstract}
 The discrete nonlinear Schr\"odinger equation is a nonlinear
 lattice system that appears in many areas of physics such as
 nonlinear optics, biomolecular chains and Bose-Einstein condensates.
 In this article, we consider a class of discrete nonlinear
 Schr\"odinger equations with unbounded potentials. We obtain some
 new sufficient conditions on the multiplicity results of ground
 state solutions for the equations by using the symmetric mountain pass
 lemma. Recent results in the literature are greatly improved.
\end{abstract}

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

\section{Introduction}
 The discrete nonlinear Schr\"odinger (DNLS) equation is one of the
 most important inherently discrete models. DNLS equations
 play a crucial role in the modeling of a great variety of phenomena,
 ranging from solid state and condensed matter physics to biology
 \cite{ChrKS,FlaG,FleSEC}. For example, they have been
 successfully applied to the modeling of localized pulse propagation
 optical fibers and wave guides, to the study of energy relaxation
 in solids, to the behavior of amorphous material, to the modeling of
 self-trapping of vibrational energy in proteins or studies related
 to the denaturation of the DNA double strand \cite{KA}.

 This article considers the DNLS equation
\begin{equation}\label{e1.1}
 i\dot{\psi}_n=-\Delta \psi_n+v_n\psi_n-\gamma_nf(\psi_n), \quad n\in \mathbb{Z},
\end{equation}
where $\Delta \psi_n=\psi_{n+1}+\psi_{n-1}-2\psi_n$ is discrete Laplacian
operator, $v_n$ and $\gamma_n$ are real valued for each $n\in \mathbb{Z}$,
$f\in C(\mathbb{R},\mathbb{R})$, $f(0)=0$ and the nonlinearity $f(u)$
is gauge invariant; that is,
\begin{equation}\label{e1.2}
 f(e^{i\theta}u)=e^{i\theta}f(u), \theta\in \mathbb{R}.
\end{equation}

 Since solitons are spatially localized time-periodic solutions and decay to zero at infinity, $\psi_n$ has the
 form
\begin{gather*}
\psi_n=u_ne^{-i\omega t}, \\
\lim_{|n|\to \infty}\psi_n=0,
\end{gather*}
 where $\psi_n$ is real valued for each $n\in \mathbb{Z}$ and 
$\omega\in\mathbb{R}$ is the temporal frequency. Then
 \eqref{e1.1} becomes
\begin{gather}\label{e1.3}
 -\Delta u_n+v_nu_n-\omega u_n=\gamma_nf(u_n), n\in \mathbb{Z}, \\
\label{e1.4}
 \lim_{|n|\to \infty}u_n=0
\end{gather}
holds, where $|n|$ is the length of index $n$. Naturally, if we look for 
solitons of \eqref{e1.1}, we just need to
 get the solutions of \eqref{e1.3} satisfying \eqref{e1.4}.

 In the past decade, the existence of solitons of the DNLS equations has 
drawn a great deal of
 interest \cite{HZ1,HZ2,MZ1,MZ2,P1,P2,Zha1,Zha2,ZhoM,ZhoY,ZhoYC}.
The existence for the periodic DNLS equations with superlinear nonlinearity 
\cite{MZ1,MZ2} and with saturable
 nonlinearity \cite{ZhoY,ZhoYC} has been studied. And the existence results
 of solitons of the DNLS equations
 without periodicity assumptions were established in \cite{HZ1,HZ2,ZhoM}. 
As for the existence of the homoclinic orbits of
 nonlinear Schr\"odinger equations, we refer to \cite{CheTi,T1,T2,T3}. 
By using the generalized Nehari
 manifold approach, Mai and Zhou \cite{MZ1} in 2013 proved the existence of 
a kind of special  solitons of \eqref{e1.3}, which called ground state 
solutions \cite{HZ2}, that is, nontrivial solutions with least possible energy 
in $l^2$. In this paper, we employ the Symmetric Mountain Pass Lemma instead of 
the generalized Nehari  manifold approach to obtain the existence of ground 
state solutions of \eqref{e1.3}.

Let $F(u)=\int_0^u f(t)dt\geq0, t\in \mathbb{R}$. Our main results are as follows.

\begin{theorem} \label{thm1.1}
 Suppose that $f(u)$ is odd in $u$ and the following hypotheses are satisfied:
 \begin{itemize}
\item[(A1)] for any $n\in\mathbb{Z}$, we have
$\underline{v}=\inf_{n\in\mathbb{Z}}v_n>\omega>0$,
 and $\lim_{|n|\to \infty} v_n=+\infty$;

\item[(A2)] there exist two positive constants $\underline{\gamma}$ and
 $\bar{\gamma}$ such that for any $n\in\mathbb{Z}$,
 $\underline{\gamma}\leq\gamma_n\leq\bar{\gamma}$;

\item[(A3)] $f(u)$ is continuous in $u$ and
$f(u)=o(u)$ as $u\to 0$;

\item[(A4)] for any $c>0$, there exist $p=p_c>0,\ q=q_c>0$ and $\mu<2$ such that
 $$
\Big(2+\frac{1}{p+q|u_n|^{\mu/2}}\Big)F(u_n)\leq f(u_n)u_n,
\quad  \forall n\in \mathbb{Z}, |u_n|\geq c;
$$

\item[(A5)] $\lim_{s\to +\infty}[\frac{\underline{\gamma}
\min_{|u|=1}F(su)}{s^2}]=+\infty.$
\end{itemize}
Then \eqref{e1.3} has an unbounded sequence solutions satisfying
\eqref{e1.4}.
\end{theorem}

\begin{theorem} \label{thm1.2}
The unbounded sequence solutions $u^{(k)}\ (k\in  \mathbb{N}$ of \eqref{e1.3} 
obtained in Theorem \ref{thm1.1} decay  exponentially at infinity:
 $$
|u^{(k)}_{n}|\leq C^{(k)}e^{-\beta^{(k)}|n|},\ n\in \mathbb{Z},
$$
 with some constants $C^{(k)}>0$ and $\beta^{(k)}>0$, $k\in \mathbb{N}$.
\end{theorem}

\begin{remark} \label{rem1.3}\rm
Zhang et al. \cite{T1,T2} studied \eqref{e1.3} under the
 assumption that 
 $$0<(q_1-1)f(u)u\leq f'(u)u^2,\ \forall u\neq0$$
 holds for some constant $q_1\in(2,+\infty)$. This is a stronger condition 
than the classical Ambrosetti- Rabinowitz superlinear condition; i.e., 
there exist constants $q_1>2$ and $r_1>0$ such that
 $$
0<q_1\int_0^uf(s)ds\leq uf(u),\ \forall |u|\geq r_1.
$$
 Thus, our results  improve the corresponding results in \cite{Zha1,Zha2}.
\end{remark}

As it is well known, critical point theory is a powerful tool to deal with the
 homoclinic solutions of differential equations
 \cite{GuAWO,GuOXA1,GuOXA2,GuOXA3} and is used to study homoclinic solutions 
of discrete systems in recent years  \cite{CheT1,CheT2,CheT3,ZhoY}. 
Our aim in this paper is to obtain the multiplicity results of ground 
state solutions for the discrete  nonlinear Schr\"odinger equations 
by using critical point theory. The main idea is to transfer the problem of solutions
in $E$ (defined in Section 2) of \eqref{e1.3} into that of critical points 
of the corresponding functional.
 The motivation for the present work stems from the recent papers 
\cite{CheT2,CheTi,GuOXA3}.


\section{Preliminaries}

To apply the critical point theory, we establish the
variational framework corresponding to \eqref{e1.3} and give 
some lemmas which will be of fundamental importance
 in proving our main results. We start by some basic notation.

Let $S$ be the vector space of all real sequences of the form
 $$
u=(\dots,u_{-n},\dots,u_{-1},u_0,u_1,\dots,u_n,
 \dots)=\{u_n\}_{n=-\infty}^{+\infty},
$$
 namely
 $S=\{\{u_n\}:u_n\in \mathbb{R},\ n\in \mathbb{Z}\}$.
Define
 $$
E=\big\{u\in S: \sum^{+\infty}_{n=-\infty}(-\Delta
 u_n\cdot u_n+v_nu_n^2)<+\infty\big\}.
$$
The space is a Hilbert space with the inner product
\begin{equation}\label{e2.1}
 \langle u,\nu\rangle =\sum^{+\infty}_{n=-\infty}(-\Delta
 u_n\cdot \nu_n+v_nu_n\nu_n), \forall u,\nu\in E,
\end{equation}
 and the corresponding norm
\begin{equation}\label{e2.2}
 \|u\|=\Big[\sum^{+\infty}_{n=-\infty}(-\Delta
 u_n\cdot u_n+v_nu_n^2)\Big]^{1/2}, \quad \forall u\in E.
\end{equation}

 In what follows, $l^2$ denotes the space of functions whose second powers are
 summable on $\mathbb{Z}$ equipped with
 $$
\|u\|_{l^2}^2=\sum_{n\in \mathbb{Z}}u_n^2.
$$
Let
 $$
l^\infty(\mathbb{Z},\mathbb{R})=\{u\in S: \sup_{n\in \mathbb{Z}}|u_n|<+\infty\}.
$$
For any $n_1,n_2\in\mathbb{Z}$ with $n_1<n_2$, we let 
$\mathbb{Z}(n_1,n_2)=[n_1,n_2]\cap\mathbb{Z}$ and
for function $f:\mathbb{Z}\to \mathbb{R}$ and $a\in\mathbb{R}$, we set
 $$
\mathbb{Z}(f_n\geq a)=\{n\in \mathbb{Z}:f_n\geq a\}, \mathbb{Z}(f_n\leq a)
=\{n\in \mathbb{Z}:f_n\leq a\}.
$$

For all $u\in E$,\ define the functional $J$ on $E$ as follows:
\begin{equation}\label{e2.3}
\begin{aligned}
J(u):=&\frac{1}{2}\sum_{n=-\infty}^{+\infty}(-\Delta
 u_n\cdot u_n+v_nu_n^2)
 -\frac{\omega}{2}\sum_{n=-\infty}^{+\infty}u_n^2
-\sum_{n=-\infty}^{+\infty}\gamma_nF(u_n) \\
=&\frac{1}{2}\|u\|^2-\frac{\omega}{2}\|u\|_{l^2}^2
 -\sum_{n=-\infty}^{+\infty}\gamma_nF(u_n).
\end{aligned}
\end{equation}

Standard arguments show that the functional $J$ is a well-defined $C^1$ 
functional on $E$ and \eqref{e1.3} is easily recognized as the corresponding
 Euler-Lagrange equation for $J$. Thus, to find nontrivial solutions 
to \eqref{e1.3} satisfying \eqref{e1.4}, we need only to look for
 nonzero critical points of $J$ in $E$.

 For the derivative of $J$ we have the formula
\begin{equation}\label{e2.4}
 \langle J'(u),\nu\rangle 
=\sum_{n=-\infty}^{+\infty}(-\Delta
 u_n\cdot \nu_n+v_nu_n\nu_n-\omega u_n\nu_n-\gamma_nf(u_n)\nu_n),\ 
\forall u,\nu\in E.
\end{equation}

Let $E$ be a real Banach space, $J\in C^1(E,\mathbb{R})$, i.e., $J$ is a
 continuously Fr\'{e}chet-differentiable functional defined on $E$. 
$J$ is said to satisfy the Palais-Smale
 condition (P.S. condition for short) if any sequence
 $\{u_n\}\subset E$ for which $\{J(u_n)\}$ is bounded and
 $J^\prime (u_n)\to  0$ $(n\to  \infty)$ possesses a
 convergent subsequence in $E$.

Let $B_\rho$ denote the open ball in $E$ about 0 of radius $\rho$
and let $\partial B_\rho$ denote its boundary.

\begin{lemma}[Symmetric Mountain Pass Lemma \cite{Ra}] \label{lem2.1}
Let  $E$ be a real Banach space and $J\in C^1(E,\mathbb{R})$ with $J$ even. 
Suppose that $J$ satisfies the P.S. condition,  $J(0)=0$,
\begin{itemize}
\item[(A6)] there exist constants $\rho, \alpha>0$ such that
 $J|_{\partial B_\rho}\geq \alpha$, and

\item[(A7)] for each finite dimensional subspace
 $\tilde{E}\subset E$, there is $r=r(\tilde{E})>0$ such that
 $J(u)\leq 0$ for $u\in\tilde{E}\backslash B_r,$ where $B_r$ 
is an open ball in $E$ of radius $r$ centered at 0.
\end{itemize}
Then $J$ possesses an unbounded sequence of critical values.
\end{lemma}

\begin{lemma} \label{lem2.2}
For $u\in E$,
\begin{equation} \label{e2.5}
\underline{v}\|u\|_\infty^2\leq\underline{v}\|u\|_{l^2}^2\leq\|u\|^2,
\end{equation}
 where
 $\|u\|_\infty=\sup_{n\in\mathbb{Z}}|u_n|$.
\end{lemma}

\begin{proof}
Since $u\in E$, it follows that
$\lim_{|n|\to \infty}|u_n|=0$. Hence, there exists
$n^*\in\mathbb{Z}$ such that
$$
\|u\|_\infty=|u_{n^*}|=\max_{n\in\mathbb{Z}}|u_n|.
$$
 By (A1) and \eqref{e2.2}, we have
 $$
\|u\|^2\geq\sum_{n\in\mathbb{Z}}v_nu_n^2
\geq\underline{v}\sum_{n\in\mathbb{Z}}u_n^2
 \geq\underline{v}\|u\|_\infty^2.
$$
 The proof is complete.
\end{proof}

\begin{lemma} \label{lem2.3}
Suppose that {\rm(A1)--(A5)}  are satisfied. Then $J$ satisfies condition {\rm (C)}
\cite{CheT2,Ra}.
\end{lemma}

\begin{proof}
Let $\{u^{(k)}\}_{k\in\mathbb{N}}\subset E$ be such that
 $\{J(u^{(k)})\}_{k\in\mathbb{N}}$ is bounded
 and $(1+\|u^{(k)}\|)\|J'(u^{(k)})\|\to  0$ as
 $k\to  \infty.$\ Then there is a positive constant $K$
 such that $|J(u^{(k)})|\leq K$. By \eqref{e2.3} and \eqref{e2.4},
 we have
\begin{equation}\label{e2.6}
\begin{aligned}
2K&\geq2J(u^{(k)})-\langle J'(u^{(k)}),u^{(k)}\rangle \\
& =\sum^{+\infty}_{n=-\infty}\gamma_n\big[f(u_n^{(k)})u_n^{(k)}-2F(u_n^{(k)})\big].
\end{aligned}
\end{equation}
 By (A3), there exists $\eta\in(0,1)$ such that
\begin{equation}\label{e2.7}
 |F(u_n)|\leq
 \frac{\underline{v}-\omega}{4\bar{\gamma}}u_n^2,\quad
 \forall n\in\mathbb{Z},\; |u_n|\leq\eta.
\end{equation}
 Then it follows from (A4) that
\begin{gather}\label{e2.8}
 f(u_n^{(k)})u_n^{(k)}>2F(u_n^{(k)})\geq0,\quad  \forall n\in\mathbb{Z}, \\
\label{e2.9}
 F(u_n^{(k)})
\leq \Big[p+q|u_n^{(k)}|^{\mu/2}\Big]
\Big[f(u_n^{(k)})u_n^{(k)} -2F(u_n^{(k)})\Big],\quad
 \forall n\in\mathbb{Z},\; |u_n^{(k)}|\geq \eta.
\end{gather}
 By Lemma \ref{lem2.2}, \eqref{e2.3}, \eqref{e2.7}, \eqref{e2.8}, \eqref{e2.10})
 and \eqref{e2.11}, we have
\begin{align*}
&\frac{1}{2}\|u^{(k)}\|^2 \\
&=J(u^{(k)})+\frac{\omega}{2}\|u^{(k)}\|_{l^2}^2
 +\sum_{n\in\mathbb{Z}(|u_n^{(k)}|\leq\eta)}\gamma_nF(u_n^{(k)})
 +\sum_{n\in\mathbb{Z}(|u_n^{(k)}|\geq\eta)}\gamma_nF(u_n^{(k)})\\
 &\leq J(u^{(k)})+\frac{\omega}{2\underline{v}}\|u^{(k)}\|^2
 +\frac{\underline{v}-\omega}{4}\sum_{n\in\mathbb{Z}(|u_n^{(k)}|\leq\eta)}
 (u_n^{(k)})^2\\
 &\quad +\bar{\gamma}\sum_{n\in\mathbb{Z}(|u_n^{(k)}|\geq\eta)}
 \big[p+q|u_n^{(k)}|^{\mu/2}\big]
 \big[f(u_n^{(k)})u_n^{(k)}-2F(u_n^{(k)})\big]\\
 &\leq K+\frac{\omega}{2\underline{v}}\|u^{(k)}\|^2
 +\frac{\underline{v}-\omega}{4\underline{v}}\|u^{(k)}\|^2
 +2K\bar{\gamma}\Big(p+q\underline{v}^{\mu/2}\|u^{(k)}\|^\mu\Big).
\end{align*}
 That is,
 $$
\frac{\underline{v}-\omega}{4\underline{v}}\|u^{(k)}\|^2
 \leq K+2K\bar{\gamma}\Big(p+q\underline{v}^{\mu/2}\|u^{(k)}\|^\mu\Big).
$$
Since $\underline{v}>\omega$ and $\mu<2$, it is not difficult to know 
that $\{u^{(k)}\}_{k\in\mathbb{N}}$
 is a bounded sequence in $E$, i.e., there exists a constant
 $K_1>0$ such that
\begin{equation}\label{e2.10}
 \|u^{(k)}\|\leq K_1,\quad k\in\mathbb{N}.
\end{equation}
So passing to a subsequence if necessary, it can be assumed that
 $u^{(k)}\rightharpoonup u^{(0)}$  in $E$. 
For any given number $\varepsilon>0$, by (A3), we can choose $\zeta>0$ such that
\begin{equation}\label{e2.11}
 |f(u)|\leq\varepsilon |u|,\quad \forall\ u\in\mathbb{R},
\end{equation}
where $|u|\leq\zeta$.

 By (A1), we can also choose a positive integer $D\in \mathbb{R}$ such that
\begin{equation}\label{e2.12}
 v_n\geq\frac{K_1^2}{\zeta^2},\quad |n|\geq D.
\end{equation}
By \eqref{e2.10} and \eqref{e2.12}, we obtain
\begin{equation}\label{e2.13}
 (u^{(k)}_n)^2=\frac{1}{v_n}v_n(u^{(k)}_n)^2
 \leq\frac{\zeta^2}{K_1^2}\|u^{(k)}\|^2\leq\zeta^2,\quad |n|\geq D.
\end{equation}
 Since $u^{(k)}\rightharpoonup u^{(0)}$ in $E$, it is easy to verify that 
$u^{(k)}_n$  converges to $u^{(0)}_n$ pointwise for all $n\in\mathbb{Z}$, that is
\begin{equation}\label{e2.14}
 \lim_{k\to \infty} u^{(k)}_n=u^{(0)}_n,\quad \forall n\in\mathbb{Z}.
\end{equation}
 Combining this with \eqref{e2.13}, we have
\begin{equation}\label{e2.15}
 (u^{(0)}_n)^2\leq\zeta^2,\quad |n|\geq D.
\end{equation}
It follows from \eqref{e2.14} and the continuity of $f(u)$ on $u$ that 
there exists $k_0\in\mathbb{N}$ such that
\begin{equation}\label{e2.16}
 \sum^{D}_{n=-D}\gamma_n|f(u^{(k)}_n)-f(u^{(0)}_n)|<\varepsilon,\quad k\geq k_0.
\end{equation}
On the other hand,  from (A3), \eqref{e2.5}, \eqref{e2.10}, \eqref{e2.11},
\eqref{e2.13}, \eqref{e2.15}  and H\"older inequality  it follows that
\begin{equation}\label{e2.17}
\begin{aligned}
 &\sum_{|n|\geq  D}\gamma_n|f(u^{(k)}_n)-f(u^{(0)}_n)||u^{(k)}_n-u^{(0)}_n|\\
 &\leq\sum_{|n|\geq D}\bar{\gamma}\big(|f(u^{(k)}_n)|+|f(u^{(0)}_n)|\Big)
 (|u^{(k)}_n|+|u^{(0)}_n|)\\
 &\leq\bar{\gamma}\varepsilon\sum_{|n|\geq D}
\big[|u^{(k)}_n|+|u^{(0)}_n|\big]
 \big(|u^{(k)}_n|+|u^{(0)}_n|\big)\\
&\leq2\bar{\gamma}\varepsilon\sum_{n=-\infty}^{+\infty}
\big(|u^{(k)}_n|^2+|u^{(0)}_n|^2\big)\\
&\leq\frac{2\bar{\gamma}\varepsilon}{\underline{v}}\big(K_1^2+\|u^{(0)}\|^2\big).
\end{aligned}
\end{equation}
Since $\varepsilon$ is arbitrary, we obtain
\begin{equation}\label{e2.18}
 \sum^{+\infty}_{n=-\infty}\gamma_n
\big|f(u^{(k)}_n)-f(u^{(0)}_n)\big|\to  0,\quad k\to \infty.
\end{equation}
It follows from \eqref{e2.2}, \eqref{e2.4} and \eqref{e2.5} that
\begin{align*}
 &\langle J'(u^{(k)})-J'(u^{(0)}),u^{(k)}-u^{(0)}\rangle \\
 &=\|u^{(k)}-u^{(0)}\|^2
 -\omega\|u^{(k)}-u^{(0)}\|_{l^2}^2
 -\sum^{+\infty}_{n=-\infty}\gamma_n(f(u^{(k)}_n)
 -f(u^{(0)}_n))
 (u^{(k)}-u^{(0)})\\
 &\geq\frac{\underline{v}-\omega}{\underline{v}}\|u^{(k)}-u^{(0)}\|^2
 -\sum^{+\infty}_{n=-\infty}\gamma_n\Big(f(u^{(k)}_n)
 -f(u^{(0)}_n)\Big)  (u^{(k)}-u^{(0)}).
\end{align*}
 Therefore, 
\begin{align*}
\frac{\underline{v}-\omega}{\underline{v}}\|u^{(k)}-u^{(0)}\|^2
 &\leq\langle J'(u^{(k)})-J'(u^{(0)}),u^{(k)}-u^{(0)}\rangle \\
&\quad +\sum^{+\infty}_{n=-\infty}\gamma_n\Big(f(u^{(k)}_n)
 -f(u^{(0)}_n)\Big)  (u^{(k)}-u^{(0)}).
\end{align*}
Since $\underline{v}>\omega>0$ and
 $\langle J'(u^{(k)})-J'(u^{(0)}),u^{(k)}-u^{(0)}\rangle \to 0, k\to \infty$.
 Thus, $u^{(k)}\to  u^{(0)}$ in $E$ and the proof is complete.
\end{proof}

\section{Proofs of theorems}

In this section, we shall prove our main results by using the
critical point method.

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 It is clear that $J$ is even and $J(0)=0$. We have already known that 
$J\in C^1(E,\mathbb{R})$ and  $J$ satisfies condition $(C)$. Hence, it
 suffices to prove that $J$ satisfies the conditions $(J_1)$
 and (A7) of Lemma \ref{lem2.1}.

If $\|u\|=\sqrt{\underline{v}}\eta:=\rho$, then by Lemma \ref{lem2.2}, 
$|u_n|\leq\eta$ for $n\in\mathbb{Z}$.
 Set $\alpha=\frac{\underline{v}-\omega}{4\underline{v}}\eta^2$. Hence,
 from \eqref{e2.3}, \eqref{e2.5}, \eqref{e2.7}, (A1) and (A3), we have
\begin{equation}\label{e3.1}
\begin{aligned}
J(u)
&\geq\frac{1}{2}\|u\|^2-\frac{\omega}{2\underline{v}}\|u\|^2
 -\sum_{n=-\infty}^{+\infty}\gamma_nF(u_n)\\
&\geq\frac{\underline{v}-\omega}{2\underline{v}}\|u\|^2
 -\frac{\underline{v}-\omega}{4\underline{v}}\sum_{n=-\infty}^{+\infty}v_nu_n^2\\
&\geq\frac{\underline{v}-\omega}{2\underline{v}}\|u\|^2
-\frac{\underline{v}-\omega}{4\underline{v}}\|u\|^2\\
 &=\frac{\underline{v}-\omega}{4\underline{v}}\|u\|^2
=\alpha.
\end{aligned} 
\end{equation}
Equation \eqref{e3.3} shows that $\|u\|=\rho$ implies that $J(u)\geq\alpha$, 
i.e., $J$ satisfies  assumption (A6).

In the following, we shall verify  condition (A7).
Let $\tilde{E}\subset E$ be a finite  dimensional subspace.
 Consider $u\in \tilde{E}$ with $u\neq0$.
 Since all norms of a finite dimensional normed space
 are equivalent, there is a constant $c_1>0$ such that
\begin{equation}\label{e3.2}
 \|u\|^2\leq c_1\|u\|^2_\infty,\quad \forall u\in\tilde{E}.
\end{equation}
Assume that dim $\tilde{E}=m$ and $u_1,u_2,\dots,u_m$ are the basis of 
$\tilde{E}$ such that for $i,j=1,2,\dots,m$, we have
\begin{equation}\label{e3.3}
 \langle u_i,u_j\rangle
 =\begin{cases}
 c_1^2,& i=j,\\
 0,& i\neq j.
 \end{cases}
\end{equation}
 Since $u_i\in E$, we can choose a positive integer $D_1>0$ such that
\begin{equation}\label{e3.4}
 |u_n^{(i)}|<\frac{1}{m},\quad |n|>D_1,\; i=1,2,\dots,m.
\end{equation}
Set $\Theta=\{u\in \tilde{E}: \|u\|=c_1\}$. Then for $u\in \Theta$, 
there exist  $\lambda_i\in\mathbb{R}$, $i=1,2,\dots,m$ such that
\begin{equation}\label{e3.5}
 u_n=\sum_{i=1}^m\lambda_iu_n^{(i)},\quad \forall n\in\mathbb{Z}.
\end{equation}
It follows that
 $$
c_1^2=\|u\|^2=\langle u,u\rangle=\sum_{i=1}^m\lambda_i^2
\langle u^{(i)},u^{(i)}\rangle =c_1^2\sum_{i=1}^m\lambda_i^2,
$$
 which implies that $|\lambda_i|\leq1$ for $i=1,2,\dots,m$. 
Hence, for $u\in\Theta$,
 let $|u_{n_0}|=\|u\|_\infty$, then by \eqref{e3.2} and \eqref{e3.4} we have
\begin{equation}\label{e3.6}
 1\leq \|u\|_\infty=|u_{n_0}|\leq\sum_{i=1}^m|\lambda_i||u_{n_0}^{(i)}|
 \leq\sum_{i=1}^m|u_{n_0}^{(i)}|,\quad \theta\in\Theta.
\end{equation}
By (A5), there exists $\sigma_0=\sigma_0(c_1, D_1)>1$ such that
\begin{equation}\label{e3.7}
 \frac{\underline{\gamma}\min_{|u|=1}F(n,su_n)}{s^2}\geq c_1^2,\quad
 \forall s\geq \sigma_0,\;  n\in\mathbb{Z}(D_1,D_1).
\end{equation}
By \eqref{e3.4} and \eqref{e3.6}, there
 exists $n_0=n_0(u)\in\mathbb{Z}(D_1,D_1)$ such that
\begin{equation}\label{e3.8}
 1\leq|u_{n_0}|=\|u\|_\infty,\quad \forall u\in\Theta.
\end{equation}
By \eqref{e2.3}, \eqref{e3.7} and \eqref{e3.8}, we have
\begin{equation}\label{e3.9}
\begin{aligned}
J(\sigma u)&=\frac{1}{2}\|\sigma u\|^2-\frac{\omega}{2}\|\sigma u\|_{l^2}^2
 -\sum_{n=-\infty}^{+\infty}\gamma_nF(\sigma u_n)\\
 &\leq\frac{\sigma^2}{2}\|u\|^2
 -\frac{\omega}{2}\|\sigma u\|_{l^2}^2-\underline{\gamma}F(\sigma u_{n_0})\\
 &\leq\frac{\sigma^2}{2}\|u\|^2-\frac{\omega}{2}\|\sigma u\|_{l^2}^2-\underline{\gamma}\min_{|x|=1}F(\sigma
 |u_{n_0}|x)\\
 &\leq\frac{(c_1\sigma)^2}{2}-(c_1\sigma)^2|u_{n_0}|^2\\
 &\leq\frac{(c_1\sigma)^2}{2}-(c_1\sigma)^2 \\
&=-\frac{(c_1\sigma)^2}{2},\quad u\in\Theta,\; \sigma\geq\sigma_0.
\end{aligned}
\end{equation}
This implies $J(u)<0$ for $u\in \tilde{E}$ and $\|u\|\geq c_1\sigma_0$. 
The condition (A7) holds. 
By Lemma \ref{lem2.1}, $J$ possesses an unbounded sequence 
$\{d^{(k)}\}_{k\in\textsc{N}}$ of critical values with
$d^{(k)}=J(u^{(k)})$, where $u^{(k)}$ is such that $J'(u^{(k)})=0$ 
for $k=1,2,\dots$. By \eqref{e2.3}, we have
$$
\frac{1}{2}\|u^{(k)}\|^2=d^{(k)}+\frac{\omega}{2}\|u^{(k)}\|_{l^2}^2
 +\sum_{n=-\infty}^{+\infty}\gamma_nF(u^{(k)}_n)\geq d^{(k)},\quad k\in\mathbb{N}.
$$
 Since $\{d^{(k)}\}_{k\in\textsc{N}}$ is unbounded, it
 follows that $\{\|u^{(k)}\|\}_{k\in\mathbb{N}}$ is unbounded.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
Similar to \cite{P1}, we can
 prove that the homoclinic solutions $u^{(k)}$ decay exponentially fast at 
infinity. For simplicity,  we omit its proof.
\end{remark}

\subsection*{Acknowledgments} 
This project is supported by the National Natural Science Foundation 
of China (No. 11401121) and Hunan Provincial Natural Science Foundation 
of China (No. 2015JJ2075).

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\end{document}