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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 273, pp. 1--13.\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.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/273\hfil Multiple homoclinic solutions]
{Multiple homoclinic solutions for indefinite second-order
 discrete Hamilton system with small perturbation}

\author[L. Zhang, X. H. Tang \hfil EJDE-2015/273\hfilneg]
{Liang Zhang, Xianhua Tang}

\address{Liang Zhang (corresponding author) \newline
School of Mathematical Sciences, University of Jinan,
Jinan 250022, China}
\email{mathspaper2012@163.com}

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

\thanks{Submitted June 9, 2015. Published October 21, 2015.}
\subjclass[2010]{39A11, 58E05, 70H05}
\keywords{Critical point; discrete Hamilton system; homoclinic solution;
\hfill\break\indent small perturbation}

\begin{abstract}
 In this article, we sutdy the multiplicity of homoclinic solutions to
 the perturbed second-order discrete Hamiltonian system
 $$
 \Delta[p(n)\Delta u(n-1)]-L(n)u(n)+\nabla W(n,u(n))+\theta\nabla F(n,u(n))=0,
 $$
 where $L(n)$ and $W(n,x)$ are neither autonomous nor periodic in $n$.
 Under the assumption that $W(n,x)$ is only locally superquardic as
 $|x|\to \infty$ and  even in $x$ and $F(n,x)$ is a perturbation term,
 we establish some existence criteria to guarantee that the above system has
 multiple homoclinic solutions by minimax method in critical point theory.
\end{abstract}

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

\section{Introduction and statement of main results}


In this article, we consider the second-order perturbed discrete Hamilton system
\begin{equation}\label{e1}
\Delta[p(n)\Delta u(n-1)]-L(n)u(n)+\nabla W(n,u(n))+\theta\nabla F(n,u(n))=0,
\end{equation}
where $n\in \mathbb{Z}$, $u\in \mathbb{R}^N$, $\Delta u(n)=u(n+1)-u(n)$ 
is the forward difference operator, $p(n)$ and $L(n)$ are $N\times N$
real symmetric positive definite matrices for all $n\in \mathbb{Z}$, and $W$, 
$F$: $\mathbb{Z}\times \mathbb{R}^{N\times N}\to \mathbb{R}$. As usual,
we say that a solution $u(n)$ of \eqref{e1} is homoclinic (to 0) if
$u(n)\to 0$ as $n\to \pm \infty$. In addition, if $u(n) \not\equiv 0$ then $u(n)$ 
is called a nontrivial homoclinic solution.

System \eqref{e1} does have its applicable setting as evidenced by the
excellent monographs (see \cite{A1,AP}), and some authors studied the existence 
of periodic solutions and subharmonic solutions of \eqref{e1}
using the critical point theory (see \cite{A2,BYG,XT,YGZ,YDG}).
 Moreover, the existence and multiplicity results of boundary value problems
for discrete inclusions, such as fourth-order discrete inclusion and partial 
difference inclusions, have been established by the application
of non-smooth version of critical point theory (see \cite{GMW,MM,MR}). 
It is obvious that system \eqref{e1} with $\theta=0$ is a discretization
of the following second-order Hamiltonian system:
\begin{equation}\label{e2}
\frac{d}{dt}(p(t)\dot{u}(t))-L(t)u(t)+\nabla W(t,u(t))=0.
\end{equation}
In recent years, the study of homoclinic solution of system \eqref{e2}
is rapid by variational methods (see \cite{CR,D,MW,OT,TL1,TX}). 
It is well known that homoclinic orbits play an important role in analyzing 
the chaos of dynamical systems. If a system has the smoothly 
connected homoclinic orbits, then it can not stand the perturbation, 
and its perturbed system probably produce chaotic phenomenon.

For system \eqref{e1} with $\theta=0$, the existence and multiplicity
of homoclinic solutions of system \eqref{e1}
or its special forms have been investigated by the use of critical point 
theory (see \cite{DCS,MG1,MG2,TL2,ZYC}). If $p(n)$, $L(n)$ and $W(n, x)$
are periodic in $n$, some authors dealt with the periodic case in \cite{DCS,MG2}. 
When the periodicity is lost, this case is quite different from the
one mentioned above, because of lack of compactness of the Sobolev embedding. 
If $W(n,x)$ is superquadratic as $|x|\to \infty$
uniformly for $n\in \mathbb{Z}$, the following well known global 
Ambrosetti-Rabinowitz superquadratic condition is often required:
\begin{itemize}
\item[(A1)] there exists $\mu>2$ such that
$$
0<\mu W(n,x)\leq (\nabla W(n,x), x), \quad
 (n,x)\in \mathbb{Z}\times \mathbb{R}^N\setminus \{0\},
$$
where and in the sequel, $(\cdot, \cdot)$ denotes the standard inner 
product in $\mathbb{R}^N$, and $|\cdot|$ is the induced norm.
\end{itemize}
However, there are many indefinite functions not satisfying (A1). For example,
let
\begin{equation}\label{e3}
W(n,x)=(n-1)|x|^s, \quad  2<s<\infty.
\end{equation}
It is obvious that $W(n,x)$ is only locally superquadratic as $|x|\to \infty$. 
If $W(n,x)$ is even in $x$, the classical multiple critical point theorems 
can be applied to obtain multiplicity results for system \eqref{e1}
with $\theta=0$. When $\theta\neq0$ and $F(n,x)$ is not even in $x$, 
then the perturbation term $F(n,x)$ breaks
the symmetry of the energy functional of system \eqref{e1}.
This case becomes different and more complicated. A natural question 
is that whether
multiple homoclinic solutions exist for system \eqref{e1} with
indefinite functions $W(n,x)$ under broken symmetry situation. 
As far as the authors are aware, there are few papers discussing this question. 
In this paper, we give a positive answer to this question. In detail, 
we obtain the following theorems.

\begin{theorem} \label{thm1.1}
Assume that $L$, $W$ and $F$ satisfy the following conditions:
\vskip2mm
\begin{itemize}
\item[(A2)] $L(n)$ is an $N\times N$ real symmetric positive definite matrix 
for all $n\in \mathbb{Z}$ and there
exists a function $l: \mathbb{Z}\to (0, \infty)$ such that 
$l(n)\to +\infty$, $|n|\to \infty$, and
$$
(L(n)x,x)\geq l(n)|x|^2, \quad  (n, x)\in \mathbb{Z}\times\mathbb{R}^N;
$$
\item[(A3)] $W(n, 0)\equiv 0$, and there exist constants $\mu>2$ such that
$$
\mu W(n,x)\leq (\nabla W(n,x), x), \quad  (n, x)\in \mathbb{Z}\times\mathbb{R}^N;
$$
\item[(A4)] for every $n\in \mathbb{Z}$, $W$ is continuously differentiable in $x$,
and there exists constants $a_1>0$ and $1<\nu_1\leq\nu_2<\infty$ such that
$$
|\nabla W(n, x)|\leq a_1 l(n)(|x|^{\nu_1}+|x|^{\nu_2}), \quad 
(n, x)\in \mathbb{Z}\times\mathbb{R}^N;
$$
\item[(A5)] there exists an infinite subset $\Lambda\subset \mathbb{Z}$ such that
$$
\lim_{|x|\to \infty}\frac{W(n,x)}{|x|^2}=\infty, \quad n\in \Lambda,
$$
and there exists $r_0\geq 0$ such that
$$
W(n,x)\geq 0, \quad  (n, x)\in \Lambda\times\mathbb{R}^N  \text{ and } 
 |x|\geq r_0;
$$
\item[(A6)] $W(n,x)=W(n,-x)$,  $(n,x)\in \mathbb{Z}\times \mathbb{R}^N$;

\item[(A7)] for every $n\in \mathbb{Z}$, $F$ is continuously differentiable 
in $x$, and there exists
a function $\gamma_1\in l^1(\mathbb{Z}, [0,+\infty))$ such that
$$
|F(n,x)|\leq \gamma_1(n), \quad  (n,x)\in \mathbb{Z}\times \mathbb{R}^N;
$$
\item[(A8)] there exists a function $\gamma_2\in l^2(\mathbb{Z}, [0,+\infty))$ 
such that
$$
|\nabla F(n,x)|\leq \gamma_2(n), \quad  (n,x)\in \mathbb{Z}\times \mathbb{R}^N.
$$
\end{itemize}
Then for any $j\in \mathbb{N}$, there exists $\varepsilon_j>0$ such that if 
$|\theta|\leq \varepsilon_j$, system \eqref{e1} possesses
at least $j$ distinct homoclinic solutions.
\end{theorem}

\begin{theorem} \label{thm1.2}
Assume that $L$, $W$ satisfy {\rm (A2)--(A6)}.
Then there exists an unbounded sequence of homoclinic solutions for system
 \eqref{e1} with $\theta=0$.
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
We would like to point out that even in the symmetric case, our results are also new.
In fact, the condition (A5) implies that $W(n, x)$ is only of locally 
superquadratic growth as $|x|\to \infty$,
and our assumption (A5) is weaker than the conditions presented in the reference.
\end{remark}

Since $F(n,x)$ is not even in $x$ in Theorem \ref{thm1.1},
the classical multiple critical point theorems fail to obtain multiplicity 
results for system \eqref{e1}.
The main difficulty is to find an appropriate class of sets due to indefinite 
character of the function $W(n, x)$ which is used to construct multiple 
critical values for the perturbed functional 
of system \eqref{e1}. To overcome this difficulty, we construct an orthogonal
sequence by which a sequence of sets are introduced, then multiple critical values 
will be obtained by minimax procedure over these sets, which correspond to
 multiple homoclinic solutions of system \eqref{e1}.

The article is organized as follows. 
In Section 2, we present some preliminary results and useful lemmas.
The proof of Theorem \ref{thm1.1} and Corollary 1.1 are given in Section 3. 
In Section 4, we present an example to illustrate our results.

 Throughout the article, we denote by $C_n$ various positive constants 
which may vary from line to line
and are not essential to the proof.


\section{Variational setting and preliminaries}

Let
\begin{gather*}
S=\big\{\{u(n)\}_{n\in \mathbb{Z}} : \ u(n)\in \mathbb{R}^N, \; n\in \mathbb{Z}\big\},
\\
E=\big\{u\in S: \ \sum_{n\in \mathbb{Z}}\Big[\big(p(n+1)\Delta u(n), 
\Delta u(n)\big)+\big(L(n)u(n), u(n)\big)\Big]<+\infty\big\}.
\end{gather*}
For $u, v\in E$, let
$$
\langle u, v\rangle=\sum_{n\in \mathbb{Z}}
\big[\big(p(n+1)\Delta u(n), \Delta v(n)\big)+\big(L(n)u(n), v(n)\big)\big].
$$
Then $E$ is a Hilbert space with the above inner product, and the corresponding 
norm is
$$
\|u\|:=\Big(\sum_{n\in \mathbb{Z}}\big[\big(p(n+1)\Delta u(n), 
\Delta u(n)\big)+\big(L(n)u(n), u(n)\big)\big]\Big)^{1/2}, \quad u\in E.
$$
Moreover, we use $E^*$ to denote the topological dual space with norm 
$\| \cdot\|_{E^*}$.
As usual, for $1\leq p<\infty$, $k=1$ or $N$, set
\begin{gather*}
l^p(\mathbb{Z}, \mathbb{R}^k)
=\big\{\{u(n)\}_{n\in \mathbb{Z}} :  u(n)\in \mathbb{R}^k, \;
 n\in \mathbb{Z}, \ \sum_{n\in \mathbb{Z}}|u(n)|^p< +\infty\big\},\\
l^\infty(\mathbb{Z}, \mathbb{R}^k)
=\big\{\{u(n)\}_{n\in \mathbb{Z}} : 
 u(n)\in \mathbb{R}^k, \; n\in \mathbb{Z}, \;
 \sup_{n\in \mathbb{Z}}|u(n)|< +\infty\big\},
\end{gather*}
and their norms are defined by
$$
\|u\|_p=\Big(\sum_{n\in \mathbb{Z}}|u(n)|^p\Big)^{1/p}, \quad
 u\in l^p(\mathbb{Z}, \mathbb{R}^k); \quad
 \|u\|_{\infty}=\sup_{n\in \mathbb{Z}}|u(n)|, \quad
 u\in l^{\infty}(\mathbb{Z}, \mathbb{R}^k).
$$
If the condition (A2) holds, $E$ is continuously embedded in
 $l^p(\mathbb{Z}, \mathbb{R}^N)$ for all $p\in [2, +\infty]$.
Consequently, there exists $\tau_p>0$ such that
\begin{equation}\label{e3b}
\|u\|_p\leq \tau_p \|u\|, \quad  u\in E.
\end{equation}

\begin{lemma} \label{lem2.1}
If condition {\rm (A2)} holds. Then $E$ is compactly embedded 
in $l^\infty(\mathbb{Z}, \mathbb{R}^N)$.
\end{lemma}

\begin{proof} 
Let $\{u_k\}$ be a bounded sequence in $E$,  that is, there is a constant 
$A$ such that
$$
\|u_k\|\leq A, \quad  k\in \mathbb{N}.
$$
Since $E$ is a reflexive space, passing to a subsequence, also denoted by 
$\{u_k\}$, it can be assumed that
$u_k\rightharpoonup u_0$, $k\to \infty$. Next we only need to prove
\begin{equation}\label{e4}
u_k\to u_0 \quad \text{in }  l^\infty(\mathbb{Z}, \mathbb{R}^N).
\end{equation}
For any given number $\varepsilon>0$, by (A2), we can choose a positive 
integer $\Pi_0$ such that
\begin{equation}\label{e5}
l(n)>\frac{4A^2}{\varepsilon^2}, \quad |n|\geq \Pi_0.
\end{equation}
By (A2) and \eqref{e5}, we have
\begin{equation}\label{e6}
|u_k(n)|^2\leq \frac{1}{l(n)}\big(L(n)u_k(n), u_k(n)\big)
\leq \frac{\varepsilon^2}{4A^2}\|u_k\|^2
\leq \frac{\varepsilon^2}{4}, \quad |n|\geq\Pi_0, \; k\in \mathbb{N}.
\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{e7}
\lim_{k\to \infty} u_k(n)= u_0(n), \quad n\in \mathbb{Z}.
\end{equation}
In view of \eqref{e6} and \eqref{e7}, we have
\begin{equation}\label{e8}
|u_0(n)|\leq \varepsilon/2,  \quad  |n|\geq\Pi_0.
\end{equation}
By \eqref{e7}, there exists $k_0\in \mathbb{N}$ such that
\begin{equation}\label{e9}
|u_k(n)-u_0(n)|\leq \varepsilon, \quad  k\geq k_0, \; |n|<\Pi_0.
\end{equation}
In combination with \eqref{e6}, \eqref{e8} and \eqref{e9}
$$
|u_k(n)-u_0(n)|\leq \varepsilon, \quad k\geq k_0, \; n\in \mathbb{Z},
$$
which implies that \eqref{e4} holds. The proof is complete.
\end{proof}

Next we introduce a functional $I: \mathbb{R}\times E \to \mathbb{R}$
\begin{equation}\label{e10}
I_{\theta}(u):=\frac{\|u\|^2}{2}-\sum_{n\in \mathbb{Z}}W(n,u(n))
-\theta\sum_{n\in \mathbb{Z}}F(n,u(n)).
\end{equation}
By (A2), (A4), (A7) and (A8), for fixed $\theta_0\in \mathbb{R}$, 
$I_{\theta_0}(u)$ is well defined and of class $C^1(E, \mathbb{R)}$. For
$u, v\in E$,
\begin{equation}\label{e11}
\begin{aligned}
\langle I'_{\theta_0}(u), v\rangle 
& =  \sum_{n\in \mathbb{Z}}\big[\big(p(n+1)\Delta u(n), \Delta v(n)\big)
 +\big(L(n)u(n), v(n)\big)\big]\\
&\quad -\sum_{n\in \mathbb{Z}}\big(\nabla W(n,u(n)), v(n)\big)
 -\theta_0\sum_{n\in \mathbb{Z}}\big(\nabla F(n,u(n)), v(n)\big).
\end{aligned}
\end{equation}
Furthermore, if $u_0\in E$ is a critical point of $I_{\theta_0}(u)$,
then $u_0$ is a homoclinic solution for system \eqref{e1}
with $\theta=\theta_0$.

\begin{lemma} \label{lem2.2}
Assume that all the hypotheses of Theorem \ref{thm1.1} hold. Then
\begin{itemize}
\item[(1)] for any fixed $\theta_0\in \mathbb{R}$, $I_{\theta_0}$ 
satisfies the Palais-Smale condition;

\item[(2)]  there exists a positive constant $C_0$ such that
$$
|I_{\theta}(u)-I_0(u)|\leq C_0|\theta|, \ \ (\theta, u)\in \mathbb{R}\times E.
$$
where $C_0:= \sum_{n\in \mathbb{Z}}|\gamma_1(n)|$.
\end{itemize}
\end{lemma}

\begin{proof} To prove (1),  we first show that there exists a constant 
$M$ such that $\{u_k\}\subset E$ is a sequence for which
\begin{equation}\label{e12}
|I_{\theta_0}(u_k)|\leq M \quad \text{and} \quad  I'_{\theta_0}(u_k)\to 0,
\end{equation}
then $\{u_k\}$ is bounded. For large $k$, it follows \eqref{e10}
and \eqref{e11} that
\begin{equation}\label{e13}
\begin{aligned}
2\mu^{-1}\|u_k\|+M
&\geq  I_{\theta_0}(u_k)-\frac{1}{\mu}\langle I'_{\theta_0}(u_k), u_k\rangle\\
&> \frac{\mu-2}{2\mu}\|u_k\|^2-C_1\|u_k\|-C_2,
\end{aligned}
\end{equation}
which implies that $\|u_k\|$ is bounded in $E$, that is, there exists a 
constant $A'>0$ such that
$$
\|u_k\|\leq A', \quad  k\in \mathbb{N}.
$$
Since $E$ is a reflexive space, passing to a subsequence, also denoted by 
$\{u_k\}$, it can be assumed that
\begin{equation}\label{e14}
u_k\rightharpoonup u_0, \quad k\to \infty.
\end{equation}
Moreover, $\|u_0\|\leq A'$ and it is easy to verify that
\begin{equation}\label{e15}
\lim_{k\to \infty} u_k(n)= u_0(n), \quad n\in \mathbb{Z}.
\end{equation}
For any given number $\varepsilon>0$, by (A4), there exists a positive constant
 $\delta<1$ such that
\begin{equation}\label{e16}
|\nabla W(n, x)|\leq \varepsilon l(n)|x|, \quad 
 (n, x)\in \mathbb{Z}\times\mathbb{R}^N, \; |x|\leq \delta.
\end{equation}
Arguing as in Lemma \ref{lem2.1}, there exists a positive integer $\Pi_0$ such that
\begin{equation}\label{e17}
|u_k(n)|\leq \delta \quad \text{and} \quad |u_0(n)|\leq \delta, \quad
 k\in \mathbb{N}, \; \ |n|> \Pi_0.
\end{equation}
It follows \eqref{e15} and the continuity of $\nabla W (n,x)$ on $x$
that there exists $k_0\in \mathbb{N}$ such that
\begin{equation}\label{e18}
\sum_{|n|\leq \Pi_0}|\nabla W(n, u_k(n))-\nabla W(n, u_0(n))||u_k(n)-u_0(n)|
<\varepsilon, \quad  k\geq k_0.
\end{equation}
On the other hand, by \eqref{e16} and \eqref{e17},
\begin{equation}\label{e19}
\begin{aligned}
&\sum_{|n|> \Pi_0}|\nabla W(n, u_k(n))-\nabla W(n, u_0(n))||u_k(n)-u_0(n)|\\
&\leq  \sum_{|n|> \Pi_0}(|\nabla W(n, u_k(n))|+|\nabla W(n, u_0(n))|)
 (|u_k(n|+|u_0(n)|)\\
&\leq  \varepsilon\sum_{|n|> \Pi_0} l(n)(|u_k(n)|+|u_0(n)|)(|u_k(n)|+|u_0(n)|)\\
&\leq  2\varepsilon\sum_{|n|> \Pi_0} l(n)(|u_k(n)|^2+|u_0(n)|^2)\\
&\leq  2\varepsilon\sum_{|n|> \Pi_0}[(L(n)u_k(n),u_k(n))+(L(n)u_0(n),u_0(n))]\\
&\leq  2\varepsilon(\|u_k\|^2+\|u_0\|^2)\\
&\leq  4\varepsilon A'^2, \quad  k\in \mathbb{N}.
\end{aligned}
\end{equation}
Since $\varepsilon$ is arbitrary, combing \eqref{e18} and \eqref{e19},
\begin{equation}\label{e20}
\sum_{n\in \mathbb{Z}}|\nabla W(n, u_k(n))-\nabla W(n, u_0(n)||u_k(n)-u_0(n)|\to 0,
 \quad k\to \infty.
\end{equation}
By (A8), there exists a positive integer $\Pi_1$ such that
\begin{equation}\label{e21}
\Big(\sum_{|n|> \Pi_1}|\gamma_2(n)|^2\Big)^{1/2}<\varepsilon.
\end{equation}
In view of (A8), \eqref{e17} and \eqref{e21}, we have
\begin{equation}\label{e22}
\begin{aligned}
&\sum_{|n|> \Pi_2}|\nabla F(n, u_k(n))-\nabla F(n, u_0(n))||u_k(n)-u_0(n)|\\
&\leq  2\Big(\sum_{|n|> \Pi_2}|\gamma_2(n)|^2\Big)^{1/2}
 \Big(\sum_{|n|> \Pi_2}|u_k(n)-u_0(n)|^2\Big)^{1/2}\\
&\leq  2\tau_2^2\|u_k-u_0\|^2\varepsilon\\
&\leq  4\tau_2^2A'^2\varepsilon, \quad k\in \mathbb{N}.
\end{aligned}
\end{equation}
where $\Pi_2:= \max\{\Pi_0, \Pi_1\}$. Moreover, it follows from the 
continuity of $\nabla F(n,x)$ on $x$ that there exists $k_1\in \mathbb{N}$ 
such that
\begin{equation}\label{e23}
\sum_{|n|\leq \Pi_2}|\nabla F(n, u_k(n))
-\nabla F(n, u_0(n))||u_k(n)-u_0(n)|<\varepsilon, \quad k\geq k_1.
\end{equation}
Since $\varepsilon$ is arbitrary, for any fixed $\theta_0\in \mathbb{R}$, 
in combination with \eqref{e17} and \eqref{e22},
\begin{equation}\label{e24}
\theta_0\sum_{n\in \mathbb{Z}}|\nabla F(n, u_k(n))
-\nabla F(n, u_0(n))||u_k(n)-u_0(n)|\to 0, \quad k\to \infty.
\end{equation}
It follows from \eqref{e12} and \eqref{e14} that
\begin{equation}\label{e25}
\langle I'_{\theta_0}(u_k)-I'_{\theta_0}(u_0), u_k-u_0\rangle:=\epsilon_k\to 0, 
\quad k\to \infty.
\end{equation}
It follows from \eqref{e20}, \eqref{e24} and \eqref{e25} that
\begin{align*}
\|u_k-u_0\|^2
&\leq \sum_{n\in \mathbb{Z}}|\nabla W(n, u_k(n))
 -\nabla W(n, u_0(n))|| u_k(n)-u_0(n)| \\
& \quad +\theta_0\sum_{n\in \mathbb{Z}}|\nabla F(n, u_k(n))
 -\nabla F(n, u_0(n))||u_k(n)-u_0(n)|+|\epsilon_k|,
\end{align*}
which implies that $u_k\to u_0$ in $E$.
Hence, $I_{\theta_0}$ satisfies Palais-Smale condition.

To prove (2), by (A7) and direct computations,
$$
|I_{\theta}(u)-I_0(u)|\leq C_0|\theta|, \ \ (\theta, u)\in \mathbb{R}\times E.
$$
The proof is complete.
\end{proof}

\begin{lemma} \label{lem2.3}
Suppose that {\rm (A5)} holds. Then there exists a normalized orthogonal 
sequence $\{\phi_i\}_{i=1}^\infty\subset E$.
\end{lemma}

\begin{proof} 
Since $\Lambda\subset \mathbb{Z}$ is an infinite set, there exist
a strictly increasing sequence or a strictly decreasing sequence
 $\{n_k\}_{k=1}^{\infty}\subset \Lambda$. Without loss of generality, we assume
$$
n_1<n_2<\dots<n_k<\dots \to +\infty.
$$
Define
\begin{equation}\label{e26}
\phi_i(n)= \begin{cases}
(1,0,\dots,0)^\top\in \mathbb{R}^N,  &n=n_i,\\
0, &n\neq n_i.
\end{cases}
\end{equation}
It is obvious that $\{\phi_i\}_{i=1}^\infty$ forms a linearly independent 
sequence in $E$. By Gram-Schmidt orthogonalization
process, also denoted by $\{\phi_i\}$, we can get a normalized orthogonal 
sequence. The proof is complete.
\end{proof}

Let $D_m=\operatorname{span}\{\phi_1, \ldots, \phi_m\}$, $m\in \mathbb{N}$. 
It is obvious that $D_m$ is a finite dimensional subspace in $E$.
Next we prove that there exists a strictly increasing sequence of numbers 
$R_m$ such that
\begin{equation}\label{e27}
I_0(u)\leq 0, \quad u\in D_m\backslash B_{R_m},
\end{equation}
where $B_{R_m}$ denotes the open ball of radius $R_m$ centered at 0 in $E$, 
and $\bar{B}_{R_m}$ denotes the closure of $B_{R_m}$ in $E$.

\begin{lemma} \label{lem2.4} 
Under assumptions {\rm (A5)}, for any finite dimensional subspace $D_m\subset E$,
\begin{equation}\label{e28}
I_0(u)\to -\infty, \quad  \|u\|\to \infty, \; u\in D_m.
\end{equation}
\end{lemma}

\begin{proof}
 We prove \eqref{e28} by contradiction. If \eqref{e28} is false,
there exists a sequence $\{u_k\}\subset D_m$ with $\|u_k\|\to \infty$, 
there exists $M>0$
such that $I_0(u_k)\geq -M$ for all $k\in \mathbb{N}$. Set $v_k=u_k/\|u_k\|$, 
then $\|v_k\|=1$. Passing to subsequence, we may assume 
$v_k\rightharpoonup v$ in $E$. Since
$D_m$ is a finite dimensional space, then $v_k\to v\in D_m$, then $\|v\|=1$. Set
$$
\Pi=\{n\in \mathbb{Z} : v(n)\neq 0\} \quad \text{and} \quad
 \Theta=\{n_1, n_2, \dots, n_m\},
$$
then
\begin{equation}\label{e29}
\Pi\neq \emptyset \quad \text{and} \quad \Pi\subset \Theta,
\end{equation}
moreover,
\begin{equation}\label{e30}
\lim_{k\to \infty}|u_k(n)|=\infty, \quad n\in \Pi.
\end{equation}
It follows from (A3) and (A4) that
\begin{equation}\label{e31}
|W(n,x)|\leq a_1l(n)(|x|^{\nu_1+1}+|x|^{\nu_2+1}), \quad
 (n, x)\in \mathbb{Z}\times\mathbb{R}^N.
\end{equation}
For $0\leq a<b$, let
\begin{equation*}
\Omega_k(a,b)=\big\{n\in \Theta: a\leq |u_k(n)|<b\big\},
\end{equation*}
it follows from \eqref{e30} that $\Pi\subset \Omega_k(r_0, \infty)$
for large $k\in \mathbb{N}$. By (A3), (A5), \eqref{e29}, \eqref{e30} and
\eqref{e31} that
\begin{align*}
0&\leq \lim_{k\to \infty}\frac{I_0(u_k)}{\|u_k\|^2}
=\lim_{k\to \infty}\Big[\frac{1}{2}-\sum_{n\in \mathbb{Z}}
 \frac{W(n, u_k)}{\|u_k\|^2}\Big]\\
&=\lim_{k\to \infty}\Big[\frac{1}{2}-\sum_{n\in \Theta}
 \frac{W(n, u_k)}{\|u_k\|^2}\Big]\\
&=  \lim_{k\to \infty}\Big[\frac{1}{2}-\sum_{n\in \Omega_k(0,r_0)}
 \frac{W(n,u_k)}{\|u_k\|^2}-\sum_{n\in\Omega_k(r_0, \infty)}
 \frac{W(n, u_k)}{|u_k|^2}|v_k|^2\Big]\\
&\leq \limsup_{k\to \infty} \Big[\frac{1}{2}+ma_ma_1\big(r_0^{\nu_1+1}
 +r_0^{\nu_2+1}\big)\|u_k\|^{-2}
 -\sum_{n\in\Omega_k(r_0, \infty)}\frac{W(n,u_k)}{|u_k|^2}|v_k|^2\Big]\\
&\leq \frac{1}{2}-\liminf_{k\to \infty}\sum_{n\in\Omega_k(r_0, \infty)}
 \frac{W(n,u_k)}{|u_k|^2}|v_k|^2
= -\infty,
\end{align*}
where $a_m:=\max\{l(n), n\in\Theta\}$. But the above inequality can not hold. 
Thus \eqref{e28} holds. The proof is complete.
\end{proof}

\section{Proofs of main results}

Next we introduce some continuous maps in $E$. Set
\begin{equation}\label{e32}
\Gamma_m=\{h\in C(F_m, E)| \ h \ \text{is odd and} \ h=\text{id} \ \text{on} \ \partial B_{R_m}\cap D_m\},
\end{equation}
where $F_m:=\bar{B}_{R_m}\cap D_m$. By \eqref{e32}, we define a sequences of minimax values
\begin{equation}\label{e33}
b_m=\inf_{h\in \Gamma_m}\max_{u\in F_m}I_0(h(u)).
\end{equation}
Since $E$ is a separable Hilbert space, there exists a total orthonormal 
basis $\{e_j\}$ of $E$. Define $X_j=\mathbb{R}e_j$, $j\in \mathbb{N}$
and
\begin{equation}\label{e34}
Y_k=\oplus_{j=1}^k X_j, \quad Z_k=\overline{\oplus_{j=k+1}^\infty X_j}, \quad
k\in \mathbb{N}.
\end{equation}
It is obvious that
$$
E=Y_k\oplus Z_k, \quad Z_k=Y_k^\bot, \quad k\in \mathbb{N}.
$$
Next we give an intersection property which has
been essentially proved by Rabinowitz in Proposition 9.23 of \cite{R2}.

\begin{lemma} \label{lem3.1} 
For any $m\in \mathbb{N}$, $\rho< R_m$ and $h\in \Gamma_m$,
$$
h(F_m)\cap \partial B_{\rho} \cap Z_{m-1}\neq \emptyset.
$$
\end{lemma}

\begin{lemma} \label{lem3.2} 
Suppose that {\rm (A2)} hold. Then
\begin{equation}\label{e35}
\beta_k:=\sup_{u\in Z_k, \quad \|u\|=1}\|u\|_{\infty}\to 0, \quad  k\to \infty.
\end{equation}
\end{lemma}

\begin{proof} 
 In fact, it is obvious that $\beta_k\geq\beta_{k+1}>0$, so 
$\beta_k\to\beta\geq0$ as $k\to \infty$. For $k\in \mathbb{N}$, 
there exists $u_k\in Z_k$ such that
\begin{equation}\label{e36}
\|u_k\|=1 \quad \text{and} \quad \|u_k\|_{\infty}>\beta_k/2.
\end{equation}
By a similar proof in \cite[Lemma 3.8]{W}, $u_k\rightharpoonup 0$ in $E$. 
By Lemma \ref{lem2.1}, we have
\begin{equation}\label{e37}
u_k\to 0 \quad  \text{in }  l^\infty(\mathbb{Z}, \mathbb{R}^N).
\end{equation}
In combination with \eqref{e36} and \eqref{e37}, \eqref{e35}
 holds. The proof is complete.
\end{proof}

\begin{lemma} \label{lem3.3}
Assume {\rm (A3)} and {\rm (A4)} hold. Then
\begin{equation}\label{e38}
b_m\to \infty, \quad m\to \infty.
\end{equation}
\end{lemma}

\begin{proof}
By Lemma \ref{lem3.1}, for any $h\in \Gamma_m$ and $\rho< R_m$, there 
exists $u_m\in h(F_m)\cap \partial B_{\rho}\cap Z_{m-1}$, then
\begin{equation}\label{e39}
\max_{u\in F_m} I_0(h(u))\geq I_0(u_m)
\geq \inf_{u\in \partial B_\rho \cap Z_{m-1}}I_0(u).
\end{equation}
In view of  (A3) and (A4),
\begin{equation}\label{e40}
|W(n,x)|\leq a_1 l(n)(|x|^{\nu_1+1}+|x|^{\nu_2+1}), \quad 
 (n, x)\in \mathbb{Z}\times\mathbb{R}^N.
\end{equation}
By (A2), \eqref{e10}, \eqref{e35} and \eqref{e40},
for $u\in Z_{m-1}$,
\begin{equation}\label{e41}
\begin{aligned}
I_0(u)
& =  \frac{\|u\|^2}{2}-\sum_{n\in \mathbb{Z}}W(n,u(n))\\
& \geq  \frac{\|u\|^2}{2}-a_1\sum_{n\in \mathbb{Z}}l(n)(|u(n)|^{\nu_1+1}
 +|u(n)|^{\nu_2+1})\\
& \geq  \frac{\|u\|^2}{2}-a_1\beta_{m-1}^{\nu_1-1}\|u\|^{\nu_1+1}
 -a_1\beta_{m-1}^{\nu_2-1}\|u\|^{\nu_2+1}.
\end{aligned}
\end{equation}
In view of \eqref{e35} and \eqref{e41}, when $m$ is large enough,
for $u\in Z_{m-1}$,
\begin{equation}\label{e42}
\begin{aligned}
I_0(u)\geq  \frac{\|u\|^2}{2}-2a_1\beta_{m-1}^{\nu_1-1}\|u\|^{\nu_2+1}-C_3.
\end{aligned}
\end{equation}
Choose $\rho:=(8a_1\beta_{m-1}^{\nu_1-1})^{\frac{1}{1-\nu_2}}$, 
if $u\in Z_{m-1}$ and $\|u\|=\rho$,
\begin{equation}\label{e43}
I_0(u)\geq \frac{1}{4}(8a_1\beta_{m-1}^{\nu_1-1})^{\frac{2}{1-\nu_2}}-C_3.
\end{equation}
In combination with \eqref{e39} and \eqref{e43}, when
$m$ is large enough,
$$
b_m\geq \frac{1}{4}(8a_1\beta_{m-1}^{\nu_1-1})^{\frac{2}{1-\nu_2}}-C_3,
$$
which implies that \eqref{e38} holds by \eqref{e35}.
The proof is complete.
\end{proof}

Next we introduce some continuous maps in $E$. Set
\begin{equation}\label{e44}
\begin{aligned}
\Lambda_m:=\big\{& H  \in C (U_m, E)| \ H|_{F_m}\in \Gamma_m \text{ and } 
  H=\text{id}   \text{ for}\\
 &u\in Q_m:=(\partial B_{R_{m+1}}\cap D_{m+1})
 \cup\big(( B_{R_{m+1}}\backslash \bar{B}_{R_m})\cap D_m\big)\big\},
\end{aligned}
\end{equation}
where
\begin{equation}\label{e45}
U_m:=\big\{u=t\phi_{m+1}+\omega: t\in [0, R_{m+1}], \;
 \omega\in \bar{B}_{R_{m+1}}\cap D_m, \; \|u\|\leq R_{m+1}\big\}.
\end{equation}

In view of Lemma \ref{lem3.3}, it is impossible that $b_{m+1}=b_m$ for all 
large $m$.
Next we can construct critical values of $I_\theta(u)$ as follows.

\begin{lemma} \label{lem3.4}
Suppose $b_{m+1}>b_m>0$. For any $\delta\in (0,  b_{m+1}-b_m)$, define
\begin{equation}\label{e46}
\Lambda_m(\delta)=\big\{H\in \Lambda_m|  I_0(H(u))\leq b_m+\delta \text{ for } 
 u\in F_m\big\}.
\end{equation}
For any $|\theta|<2C_0^{-1}(b_{m+1}-b_m-\delta)$, where $C_0$ is given in 
Lemma \ref{lem2.2}, let
\begin{equation}\label{e47}
c_m(\theta)=\inf_{H\in \Lambda_m(\delta)}\max_{u\in U_m}I_{\theta}(H(u)).
\end{equation}
Then $c_m(\theta)$ is a critical value of $I_{\theta}(u)$.
\end{lemma}

\begin{proof} 
By (2) in Lemma \ref{lem2.2}, we have
\begin{equation}\label{e48}
I_0(u)-C_0|\theta|\leq I_{\theta}(u)\leq I_0(u)+C_0|\theta|, \quad
 (\theta,u)\in \mathbb{R}\times E.
\end{equation}
For any $H\in \Lambda_m(\delta)$, since $F_{m+1}=U_m \cup
(-U_m)$, then $H$ can be continuously extended to $F_{m+1}$ as an odd 
function $\bar{H}$. Moreover,
$\bar{H}\in \Gamma_{m+1}$. Since $I_0(u)$ is even, by the construction of 
$\bar{H}$, we have
\begin{equation}\label{e49}
\max_{x\in U_m} I_0(H(x))=\max_{x\in F_{m+1}} I_0(\bar{H}(x)).
\end{equation}
It follows from \eqref{e33}, \eqref{e48} and \eqref{e49} that
\begin{equation}\label{e50}
\begin{aligned}
\max_{x\in U_m}I_{\theta}(H(x))
&\geq  \max_{x\in U_m}I_0(H(x))-C_0|\theta|\\
&=  \max_{x\in F_{m+1}}I_0(\bar{H}(x))-C_0|\theta|\\
&\geq  b_{m+1}-C_0|\theta|.
\end{aligned}
\end{equation}
In view of \eqref{e47} and \eqref{e50}, we obtain
\begin{equation}\label{e51}
c_m(\theta)\geq b_{m+1}-C_0|\theta|> b_m+\delta+C_0|\theta|.
\end{equation}
If we choose $H_m\in \Lambda_m(\delta)$, then $H_m$ can be continuously 
extended to $F_{m+1}$ as an odd function $\bar{H}_m$.
Moreover, $\bar{H}_m\in \Gamma_{m+1}$. Define
\begin{equation}\label{e52}
c_m=\max_{x\in U_m}I_0(H_m(x)).
\end{equation}
It is obvious that $c_m<+\infty$ and $c_m$ is independent of $\theta$. 
It follows from \eqref{e33} and \eqref{e52} that
\begin{equation}\label{e53}
c_{m}=\max_{x\in U_m}I_0(H_m(x))=\max_{x\in F_{m+1}} I_0(\bar{H}_m(x))\geq b_{m+1}.
\end{equation}
Moreover, by \eqref{e47}, \eqref{e48} and \eqref{e52},
\begin{equation}\label{e54}
c_m(\theta)\leq c_m+C_0|\theta|.
\end{equation}
Next we show that $c_m(\theta)$ is a critical value of $I_{\theta}(u)$. 
If $c_m(\theta)$ is a regular value of $I_{\theta}(u)$,
by \eqref{e51}, choose
\begin{equation}\label{e55}
\bar{\varepsilon}=(c_m(\theta)-b_m-\delta-C_0|\theta|)/2,
\end{equation}
By the Deformation Theorem in \cite{R2},
there exists $\varepsilon\in (0, \bar{\varepsilon})$ and 
$\eta\in C([0,1]\times E, E)$ such that
\begin{equation}\label{e56}
\eta(1,u)=u, \quad I_{\theta}(u)\not\in [c_{m}(\theta)
-\bar\varepsilon,c_{m}(\theta)+\bar\varepsilon],
\end{equation}
and if $I_{\theta}(u)\leq c_{m}(\theta)+\varepsilon$, then
\begin{equation}\label{e57}
I_{\theta}(\eta(1, u))\leq c_{m}(\theta)-\varepsilon.
\end{equation}
By \eqref{e47}, there exists
$H_0\in\Lambda_{m}(\delta)$ such that
\begin{equation}\label{e58}
\max_{u\in U_m}I_{\theta}(H_0(u))<c_{m}(\theta)+\varepsilon.
\end{equation}
Define
\begin{equation}\label{e59}
\bar H_0(\cdot)=\eta(1, H_0(\cdot)).
\end{equation}
Next we prove $\bar H_0\in \Lambda_{m}(\delta)$. 
It is obvious that $\bar H_0\in C(U_m, E)$. In view of 
$H_0\in \Lambda_m(\delta)$, \eqref{e46}, \eqref{e48}
and \eqref{e55},
\begin{equation}\label{e60}
I_{\theta}(H_0(u))\leq I_0(H_0(u))+C_0|\theta|
\leq b_m+\delta+C_0|\theta|< c_{m}(\theta)-\bar{\varepsilon}, \quad \ u\in F_m.
\end{equation}
In combination with \eqref{e56}, \eqref{e59} and \eqref{e60},
$$
\bar H_0(u)=\eta(1, H_0(u))=H_0(u), \quad u\in  F_m,
$$
which yields 
\begin{equation}\label{e61}
\bar H_0|_{F_m}\in \Gamma_m \quad \text{and} \quad 
 I_0(\bar H_0(u))=I_0(H_0(u))\leq b_m+\delta, \; u\in F_m.
\end{equation}
In view of $H_0\in \Lambda_{m}(\delta)$ and the definitions of 
$R_m$ and $R_{m+1}$
\begin{equation}\label{e62}
H_0(u)=u \quad \text{and} \quad I_0(H_0(u))\leq 0, \quad u\in Q_m.
\end{equation}
By \eqref{e48}, \eqref{e55} and \eqref{e62}, we have
\begin{equation}\label{e63}
I_{\theta}( H_0(u))\leq I_0(H_0(u))+C_0|\theta|
\leq C_0|\theta|< c_{m}(\theta)-\bar{\varepsilon}, \quad u\in Q_m.
\end{equation}
It follows \eqref{e56}, \eqref{e62} and \eqref{e63} that
\begin{equation}\label{e64}
\bar H_0(u)=\eta(1, H_0(u))=H_0(u)=u, \quad u\in  Q_m.
\end{equation}
In view of \eqref{e61} and \eqref{e64},
$\bar H_0\in \Lambda_m(\delta)$. Moreover, it follows \eqref{e57}
and \eqref{e58} that
$$
\max_{u\in U_m}I_\theta\big(\bar{H}_0(u)\big)
=\max_{u\in U_m}I_\theta\big(\eta(1,H_0(u))\big)\leq c_{m}(\theta)-\varepsilon,
$$
which is a contradiction to \eqref{e47}. The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
First we can choose a subsequence $\{n_k\}\subset \mathbb{N}$ such that 
$b_{n_k+1}>b_{n_k}> 0$. In view of Lemma \ref{lem3.4},
there exist two sequences $\{\theta_k\}$ and $\{c_{n_k}(\theta)\}$ 
such that $\theta_k>0$ and $c_{n_k}(\theta)$ is a critical value for 
$I_{\theta}(u)$
with $|\theta|\leq \theta_k$. Moreover, by \eqref{e51} and \eqref{e54},
\begin{equation}\label{e65}
b_{n_k}-C_0|\theta|\leq c_{n_k}(\theta)\leq c_{n_k}+C_0|\theta|.
\end{equation}
For any $j\in \mathbb{N}$, choose strictly increasing integers $p_i$ 
such that for $1\leq i \leq j$,
$$
p_i\in \{n_k\}\quad \text{and} \quad c_{p_i}<b_{p_{(i+1)}}.
$$
Next we can choose $\varepsilon_j>0$ small enough such that 
$c_{p_i}(\theta)$ with $1\leq i \leq j$ are defined for 
$|\theta|\leq \varepsilon_j$. Moreover, if $|\theta|\leq \varepsilon_j$, 
for $1\leq i \leq j$,
\begin{equation}\label{e66}
c_{p_i}+C_0|\theta|<b_{p_{(i+1)}}-C_0|\theta|.
\end{equation}
In view of \eqref{e65} and \eqref{e66}, for
$|\theta|\leq \varepsilon_j$, $I_\theta(u)$ has at least $j$ critical values and
$$
c_{p_1}(\theta)<c_{p_2}(\theta)<\dots<c_{p_j}(\theta).
$$
Therefore system \eqref{e1} has at least $j$ distinct solutions.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
By the Deformation Theorem and  Lemma \ref{lem3.4}, we can prove that
 $\{b_m\}$ is a sequence of critical values of $I_0(u)$ which converge to
$+\infty$. Hence the corresponding critical points are solutions of 
system \eqref{e1} with $\theta=0$. The proof is complete.
\end{proof}

\section{Examples}
In this section, we give an example to illustrate our results.
In system \eqref{e1}, let $p(n)$ be an $N\times N$ real symmetric
positive definite matrix
for all $n\in \mathbb{Z}$, $L(n)=(n^2+1)I_N$, and let
$$
W(n, x)=(n^2-10)|x|^3 \quad \text{and} \quad  F(n,x)=\frac{\sin x_1}{1+n^2},
$$
where $x=(x_1, x_2, \dots x_N)$. Thus all conditions of Theorem \ref{thm1.1}
 are satisfied with
$$
\mu=3, \quad \nu_1=\nu_2=2, \quad \gamma_1(n)=\gamma_2(n)=\frac{1}{1+n^2}.
$$
By Theorem \ref{thm1.1}, for any $j\in \mathbb{N}$, there exists $\varepsilon_j>0$ 
such that if $|\theta|\leq \varepsilon_j$, then system \eqref{e1} possesses
at least $j$ distinct solutions. Since $F(n, x)$ in our example is not
 even in $x$, the results in \cite{DCS,MG1,MG2,TL2,ZYC} can't be applied to 
this example.

\subsection*{Acknowledgments}
This research was supported by the
National Natural Science Foundation of China (No. 11171351, 11571370),
the NSF of Shandong Province of China (No. ZR2014AP011).


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\end{document}