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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 251, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/251\hfil Existence of infinitely many periodic solutions]
{Existence of infinitely many periodic solutions for
second-order Hamiltonian systems}

\author[H. Gu, T. An \hfil EJDE-2013/251\hfilneg]
{Hua Gu, Tianqing An}  % in alphabetical order

\address{Hua Gu \newline
College of Science, Hohai University, Nanjing 210098, China}
\email{guhuasy@hhu.edu.cn}

\address{Tianqing An  \newline
College of Science, Hohai University, Nanjing 210098, China}
\email{antq@hhu.edu.cn}

\thanks{Submitted June 21, 2013. Published November 20, 2013.}
\subjclass[2000]{34C25, 58E05}
\keywords{Periodic solution; Hamiltonian systems; critical point; 
\hfill\break\indent variational method}

\begin{abstract}
 By using the variant of the fountain theorem, we study the existence of
 infinitely many periodic solutions for a class of superquadratic
 nonautonomous second-order Hamiltonian systems.
\end{abstract}

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


\section{Introduction}\label{s:1}

Consider the second-order Hamiltonian system
\begin{equation}\label{eq:1.1}
\begin{gathered}
\ddot{u}(t)-U(t)u(t)+\nabla_{u}W(t,u)=0,\quad \forall t\in \mathbb{R},\\
u(0)=u(T),\quad \dot{u}(0)=\dot{u}(T),\quad T>0,
\end{gathered}
\end{equation}
where $U(\cdot)$ is a continuous $T$-periodic symmetric
positive definite matrix and $W:
\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}$ is
$T$-periodic in its first variable. Moreover, we  assume
that $W(t,x)$ is continuous in $t$ for each $x\in \mathbb{R}^{N}$,
continuously differentiable in $x$ for each $t\in[0,T]$ and $\nabla
W(t,x)$ denotes its gradient with respect to the $x$ variable.

Inspired by the monographs \cite{5,6}, the existence and
multiplicity of periodic solutions for Hamiltonian systems have been
investigated in many papers (see \cite{2,3,4,7,8,9,10,12}
 and the references therein) via the
variational methods. In 2008, He and Wu \cite{4} studied the
existence of nontrivial $T$-periodic solutions for system
\eqref{eq:1.1} by a mountain pass theorem and a local link theorem.
In 2010, Zhang and Tang \cite{10}  obtained some new results of
$T$-periodic solutions for system \eqref{eq:1.1} under weaker
assumptions, which generalized the corresponding results in
\cite{4}. In \cite{9}, Zhang and Liu considered the
second-order Hamiltonian system
\begin{equation}\label{eq:1.2}
\begin{gathered}
\ddot{u}(t)+\nabla_{u}V(t,u)=0,\quad \forall t\in \mathbb{R},\\
u(0)=u(T), \quad \dot{u}(0)=\dot{u}(T),\quad T>0,
\end{gathered}
\end{equation}
where $V\in C^{1}(\mathbb{R}\times \mathbb{R}^{N})$ is
$T$-periodic in $t$ and has the form
\begin{equation*}
V(t,u)=\frac{1}{2}\langle U(t)u,u\rangle+W(t,u).
\end{equation*}
Here and in the sequel, $\langle\cdot, \cdot\rangle$ and $|\cdot|$
denote the standard inner product and  norm in
$\mathbb{R}^{N}$ respectively. They obtained
infinitely many periodic solutions of \eqref{eq:1.2} by using the
variant of the fountain theorem under  superquadratic
assumptions (see \cite[Theorem 1.2]{9}).

Now motivated by the above papers \cite{9,10}, we will use the
following conditions to obtain the existence of infinitely many
periodic solutions of system \eqref{eq:1.1}.
\begin{itemize}
  \item[(S1)] There exist constants $d_1>0$ and $\alpha>1$ such
  that
  $$
|\nabla_{u}W(t,u)| \leq d_1(1+|u|^{\alpha}),\quad
 \forall t\in[0,T],\; u\in\mathbb{R}^{N};
$$

  \item[(S2)] $W(t,u)\geq 0$ for all $(t,u) \in [0,T]\times\mathbb{R}^{N}$,
  and
  $$
\liminf_{|u|\to\infty}\frac{W(t,u)}{|u|^2}=\infty,\quad \forall t\in [0,T];
$$

   \item[(S3)] There exist constants $\mu>2$, $0<\beta<2$, $L>0$
  and a function $a(t)\in L^{1}(0,T;R^{+})$ such that
  $$
\mu W(t,u)\leq \langle\nabla_{u}W(t,u),u\rangle+a(t)|u|^{\beta},\quad
\forall |u|\geq L,\; u\in\mathbb{R}^{N},\; t\in[0,T].
$$
\end{itemize}
Then our main result is the following theorem.

\begin{theorem}\label{thm:1.1}
Assume that {\rm (S1)--(S3)}  hold and  that $W(t,u)$ is
even in $u$. Then \eqref{eq:1.1} possesses infinitely many
solutions.
\end{theorem}

Note that by (S1), we can obtain that there
exists a constant $d_2>0$ such that
\begin{equation}\label{eq:1.3}
|W(t,u)|\leq d_1(|u|+|u|^{\alpha+1})+d_2,\quad
\forall(t,u)\in[0,T]\times\mathbb{R}^{N}.
\end{equation}
As is known, the so-called global Ambrosetti-Rabinowitz condition
(AR-condition for short) was introduced by Ambrosetti and Rabinowitz
in \cite{1} and is wildly used in the study of the
superquadratic case of Hamiltonian systems: there is a constant
$\mu$ such that
\begin{equation}\label{eq:1.4}
0<\mu W(t,u)\leq \langle\nabla_{u}W(t,u),u\rangle,\quad  \forall
(t,u)\in\mathbb{R}\times\mathbb{R}^{N}\setminus\{0\}.
\end{equation}
When we take $a(t)\equiv0$, the condition (S3) reduces to
\eqref{eq:1.4}. So the condition (S3) is weaker than
AR-condition.


\section{Preliminaries}\label{s:2}

In this section, we will establish the variational setting for our
problem and give a variant fountain theorem. Let $E=H_{T}^{1}$ be
the usual Sobolev space with the inner product
\begin{equation*}
\langle u,v\rangle_{E}
=\int_0^{T}\langle u(t),v(t)\rangle dt
+\int_0^{T}\langle\dot{u}(t),\dot{v}(t)\rangle dt.
\end{equation*}
We define a functional $\Phi$ on $E$ by
\begin{equation}\label{eq:2.1}
\Phi(u)=\frac{1}{2}\Big(\int_0^{T}|\dot{u}|^2dt+\int_0^{T}\langle
U(t)u,u\rangle dt\Big)-\Psi(u),
\end{equation}
where $\Psi(u)=\int_0^{T}W(t,u(t))dt$. Then  $\Phi$ and $\Psi$ are
continuously differentiable and
\[
\langle\Phi'(u),v\rangle
=\int_0^{T}\langle\dot{u},\dot{v}\rangle dt
 +\int_0^{T}\langle U(t)u,v\rangle dt
 -\int_0^{T}\langle\nabla_{u}W(t,u),v\rangle dt.
\]

Define a selfadjoint linear operator
$\mathcal{B}: L^2([0,T],\mathbb{R}^{N})\to L^2([0,T],\mathbb{R}^{N})$ by
$$
\langle\mathcal{B}u, v\rangle_{L^2}
=\int_0^T\langle \dot{u}, \dot{v} \rangle dt
+\int_0^T\langle U(t) u, v \rangle dt
$$
with domain $D(\mathcal{B})=E$. Then $\mathcal{B}$ has a
sequence of eigenvalues
\begin{equation*}
\lambda_{-m}\leq\lambda_{-m+1}\leq\dots\leq\lambda_{-1}<
0<\lambda_1\leq\lambda_2\leq\dots\leq\lambda_{k}\leq\dots
\end{equation*}
such that $\lambda_{k}\to +\infty$ as $k\to
+\infty$. Note that 0 may be not a eigenvalue. Let $e_j$ be the
eigenvector of $\mathcal{B}$ corresponding to $\lambda_j$ (esp.
$e_0$ is an eigenvector corresponding to the eigenvalue 0.), then
$\{e_{-m},e_{-m+1},\dots,e_{-1},e_0,e_1,\dots\}$ forms an
orthogonal basis in $L^2$. Set
\begin{gather*}
E^0=\ker\mathcal{B}=\operatorname{span}\{e_0\},\\
E^{-}=\operatorname{span}\{e_j:j=-m,-m+1,\dots,-1\}, \\
E^{+}=\operatorname{span}\{e_j:j=1,2,\dots,\}.
\end{gather*}
Then $E$ possess an orthogonal decomposition $E=E^{-}\oplus
E^0\oplus E^{+}$. For $u\in E$, we have
$$
u=u^{-}+u^0+u^{+}\in E^{-}\oplus E^0\oplus E^{+}.
$$
We can define on $E$ a new inner product and the associated
norm by
\begin{gather*}
\langle u,v\rangle_0
=\langle\mathcal{B}u^{+},v^{+}\rangle_{L^2}
-\langle\mathcal{B}u^{-},v^{-}\rangle_{L^2}
+\langle u^0,v^0\rangle_{L^2},
\\
\|u\|=\langle u,u\rangle_0^{1/2}.
\end{gather*}
Therefore, $\Phi$ can be written as
\begin{equation}\label{eq:2.2}
\Phi(u)=\frac{1}{2}(\|u^{+}\|^2-\|u^{-}\|^2)-\Psi(u).
\end{equation}


Direct computations show that
\begin{equation} \label{eq:2.3}
\begin{gathered}
\langle\Psi'(u),
v\rangle = \int_0^{T}\langle\nabla_{u}W(t,u),v\rangle dt \\
\langle\Phi'(u), v\rangle = \langle u^{+},v^{+}\rangle_0-\langle
u^{-},v^{-}\rangle_0-\langle\Psi'(u), v\rangle
\end{gathered}
\end{equation}
for all $u,v\in E$ with $u=u^{-}+u^0+u^{+}$ and
$v=v^{-}+v^0+v^{+}$ respectively. It is known that
$\Psi':E\to E$ is compact.

Denote by $|\cdot|_{p}$ the usual norm of $L^{p}\equiv
L^{p}([0,T],\mathbb{R}^{N})$ for all $1\leq p\leq\infty$, then by
the Sobolev embedding theorem, there exists a $\tau_{p}>0$ such that
\begin{equation}\label{eq:2.4}
|u|_{p}\leq\tau_{p}\|u\|,\quad  \forall u\in E.
\end{equation}\

Now we state an abstract critical point theorem founded in
\cite{11}. Let $E$ be a Banach space with the norm $\|\cdot\|$
and $E=\overline{\oplus_{j\in\mathbb{ N}}X_j}$ with
$\dim X_j<\infty$ for any $j\in\mathbb{ N}$. Set
$Y_{k}=\oplus_{j=1}^{k}X_j$ and
$Z_{k}=\overline{\oplus_{j=k}^{\infty}X_j}$. Consider the
following $C^{1}$-functional $\Phi_{\lambda}:E\to\mathbb{R}$ defined by
\begin{equation*}
\Phi_{\lambda}(u):=A(u)-\lambda B(u),\quad \lambda\in [1,2].
\end{equation*}

\begin{theorem}[{\cite[Theorem 2.1]{11}}] \label{thm:2.1}
 Assume that the functional $\Phi_{\lambda}$ defined above satisfies
\begin{itemize}
\item[(F1)] $\Phi_{\lambda}$ maps bounded sets to bounded sets for
 $\lambda\in[1,2]$, and
  $\Phi_{\lambda}(-u)=\Phi_{\lambda}(u)$ for all $(\lambda,u)\in[1,2]\times
  E$;

\item[(F2)] $B(u)\geq0$ for all $u\in E$; moreover, $A(u)\to\infty$ or
 $B(u)\to\infty$ as $\|u\|\to\infty$;

\item[(F3)] There exist $r_{k}>\rho_{k}>0$ such that
\[
  \alpha_{k}(\lambda):=\inf_{u\in
  Z_{k},\|u\|=\rho_{k}}\Phi_{\lambda}(u)>\beta_{k}(\lambda):=\max_{u\in
  Y_{k},\|u\|=r_{k}}\Phi_{\lambda}(u),\quad  \forall \lambda\in[1,2].
\]
\end{itemize}
Then
\begin{equation*}
\alpha_{k}(\lambda)\leq\zeta_{k}(\lambda):=\inf_{\gamma\in\Gamma_{k}}\max_{u\in
B_{k}}\Phi_{\lambda}(\gamma(u)),\quad  \forall\lambda\in[1,2],
\end{equation*}
where $B_{k}=\{u\in Y_{k}:\|u\|\leq r_{k}\}$ and
$\Gamma_{k}:=\{\gamma\in C(B_{k},E):\gamma\text{is odd},\;
\gamma|_{\partial B_{k}}=\text{id}\}$.
Moreover, for a.e.
$\lambda\in[1,2]$, there exists a sequence
$\{u_{m}^{k}(\lambda)\}_{m=1}^{\infty}$ such that
\[
\sup_{m}\|u_{m}^{k}(\lambda)\|<\infty,\quad
\Phi'_{\lambda}(u_{m}^{k}(\lambda))\to0,\quad
\Phi_{\lambda}(u_{m}^{k}(\lambda))\to\zeta_{k}(\lambda)\quad
\text{as } m\to\infty.
\]
\end{theorem}

To apply this theorem to prove our main result, we define
the functionals \emph{A}, \emph{B} and $\Phi_{\lambda}$ on our
working space $E$ by
\begin{gather}\label{eq:2.5}
A(u)=\frac{1}{2}\|u^{+}\|^2,\quad
B(u)=\frac{1}{2}\|u^{-}\|^2+\int_0^{T}W(t,u)dt,\\
\label{eq:2.6}
\Phi_{\lambda}(u)=A(u)-\lambda
B(u)=\frac{1}{2}\|u^{+}\|^2
-\lambda(\frac{1}{2}\|u^{-}\|^2+\int_0^{T}W(t,u)dt)
\end{gather}
for all $u=u^{-}+u^0+u^{+}\in E=E^{-}+E^0+E^{+}$ and
$\lambda\in[1,2]$. Then $\Phi_{\lambda}\in C^{1}(E,\mathbb{R})$ for
all $\lambda\in[1,2]$ and
\begin{equation}\label{eq:2.7}
\langle\Phi'_{\lambda}(u), v\rangle=\langle
u^{+},v^{+}\rangle_0-\lambda(\langle
u^{-},v^{-}\rangle_0+\int_0^{T}\langle\nabla_{u}W(t,u),v\rangle
dt).
\end{equation}
 Let $X_j=\operatorname{span}\{e_j\}$,
$j=-m,-m+1,\dots,-1,0,1,2,\dots$.
Note that $\Phi_1=\Phi$, where $\Phi$ is the functional defined in
\eqref{eq:2.2}.

\section{Proof of Theorem \ref{thm:1.1}}\label{s:3}

We  first establish the following lemmas and then give the proof
of Theorem \ref{thm:1.1}.

\begin{lemma}\label{Lemma:3.1}
Assume that {\rm (S1)--(S2)} hold. Then $B(u)\geq0$ for all
$u\in E$. Furthermore, $A(u)\to\infty$ or
$B(u)\to\infty$ as $\|u\|\to\infty$.
\end{lemma}

\begin{proof}
 Since $W(t,u) \geq 0$, by \eqref{eq:2.5}, it is obvious
that $B(u)\geq0$ for all $u\in E$.
By the proof of \cite[Lemma 2.6]{9}, for any finite-dimensional
subspace $F\subset E$, there exists a constant $\epsilon>0$ such
that
\begin{equation}\label{eq:3.1}
m(\{t\in[0,T]:|u|\geq\epsilon\|u\|\})\geq\epsilon,\quad \forall u\in
F\setminus\{0\},
\end{equation}
where $m(\cdot)$ is the Lebesgue measure.

Now for the finite-dimensional subspace
$E^{-}\oplus E^0\subset E$, there exist a constant $\epsilon$
corresponding to the one in \eqref{eq:3.1}. Let
\begin{equation*}
\Lambda_{u}=\{t\in[0,T]:|u|\geq\epsilon\|u\|\},\quad  \forall u\in
E^{-}\oplus E^0\setminus\{0\}.
\end{equation*}
Then $m(\Lambda_{u})\geq\epsilon$. By (S2), there exist
positive constants $d_3$ and $R_1$ such that
\begin{equation}\label{eq:3.2}
W(t,u)\geq d_3|u|^2, \quad \ \forall t\in[0,T]\ and\ |u|\geq
R_1.
\end{equation}
Note that
\begin{equation}\label{eq:3.3}
|u(t)|\geq R_1, \quad \ \forall t\in\Lambda_{u}
\end{equation}
for any $u\in E^{-}\oplus E^0$ with $\|u\|\geq R_1/\epsilon$.
Combining \eqref{eq:3.2} and \eqref{eq:3.3}, for any
$u\in E^{-}\oplus E^0$ with $\|u\|\geq R_1/\epsilon$, we have
\begin{align*}
B(u)&=\frac{1}{2}\|u^{-}\|^2+\int_0^{T}W(t,u)dt\\
&\geq\int_{\Lambda_{u}}W(t,u)dt\geq\int_{\Lambda_{u}}d_3|u|^2dt\\
&\geq d_3\epsilon^2\|u\|^2\cdot m(\Lambda_{u})
\geq d_3\epsilon^{3}\|u\|^2\,.
\end{align*}
This implies
$B(u)\to\infty$ as $\|u\|\to\infty$ on $E^{-}\oplus E^0$.
Combining this with $E=E^{-}\oplus E^0\oplus E^{+}$ and
\eqref{eq:2.5}, we have
$$
A(u)\to\infty\text{ or }B(u)\to\infty\quad \text{as }\|u\|\to\infty.
$$
The proof is complete.
\end{proof}

\begin{lemma}\label{Lemma:3.2}
Let {\rm (S1)--(S3)} be satisfied. Then there exist a positive
integer $k_1$ and two sequences $r_{k}>\rho_{k}\to\infty$
as $k\to\infty$ such that
\begin{equation}\label{eq:3.4}
  \alpha_{k}(\lambda):=\inf_{u\in
  Z_{k},\|u\|=\rho_{k}}\Phi_{\lambda}(u)>0,\quad
  \forall k\geq k_1,
\end{equation}
and
\begin{equation}\label{eq:3.5}
  \beta_{k}(\lambda):=\max_{u\in
  Y_{k},\|u\|=r_{k}}\Phi_{\lambda}(u)<0,\quad
  \forall k\in \mathbb{N},
\end{equation}
where
$Y_{k}=\oplus_{j=-m}^{k}X_j=\operatorname{span}\{e_{-m},e_{-m+1},\dots,e_{k}\}$
and
\[
Z_{k}=\overline{\oplus_{j=k}^{\infty}X_j}=\overline{\operatorname{span}
\{e_{k},e_{k+1},\dots\}}
\]
for all $k\in\{-m,-m+1,\dots,1,2,\dots\}$.
\end{lemma}

\begin{proof} \emph{Step 1.} First we prove \eqref{eq:3.4}.
By  \eqref{eq:1.3} and \eqref{eq:2.6},  for all
$u\in E^{+}$ we have
\begin{equation} \label{eq:3.6}
\begin{aligned}
\Phi_{\lambda}(u)
&\geq \frac{1}{2}\|u\|^2-2\int_0^{T}W(t,u)dt \\
&\geq \frac{1}{2}\|u\|^2-2d_1(|u|_1+|u|_{\alpha+1}^{\alpha+1})-2d_2T,
\quad \forall\lambda\in[1,2].
\end{aligned}
\end{equation}
where $d_1$, $d_2$ are the constants in \eqref{eq:1.3}. Let
\begin{equation}\label{eq:3.7}
\iota_{\alpha+1}(k)=\sup_{u\in Z_{k},\|u\|=1}|u|_{\alpha+1},\quad
\forall k\in \mathbb{N}.
\end{equation}
Then
\begin{equation}\label{eq:3.8}
\iota_{\alpha+1}(k)\to0\quad \text{as } k\to\infty
\end{equation}
since $E$ is compactly embedded into $L^{\alpha+1}$. Note that
\begin{equation}\label{eq:3.9}
Z_{k}\subset E^{+},\quad \forall k\geq1.
\end{equation}
Combining \eqref{eq:2.4}, \eqref{eq:3.6}, \eqref{eq:3.7} and
\eqref{eq:3.9}, for $k\geq1$, we have
\begin{equation}\label{eq:3.10}
\Phi_{\lambda}(u)\geq\frac{1}{2}\|u\|^2-2d_1\tau_1\|u\|
-2d_2T-2d_1\iota_{\alpha+1}^{\alpha+1}(k)\|u\|^{\alpha+1},
\end{equation}
for all $(\lambda,u)\in [1,2]\times Z_{k}$,
where $\tau_1$ is the constant given in \eqref{eq:2.4}. By
\eqref{eq:3.8}, there exists a positive integer $k_1\geq1$ such
that
\begin{equation}\label{eq:3.11}
\rho_{k}:=(16d_1\iota_{\alpha+1}^{\alpha+1}(k))^{1/(1-\alpha)}
>\max\{16d_1\tau_1+1,16d_2T\},\quad  \forall k\geq k_1
\end{equation}
since $\alpha>1$. Clearly,
\begin{equation}\label{eq:3.12}
\rho_{k}\to\infty\quad \ as\ k\to\infty.
\end{equation}
Combining \eqref{eq:3.10} and \eqref{eq:3.11}, direct computation
shows
\begin{equation*}
 \alpha_{k}(\lambda):=\inf_{u\in
  Z_{k},\|u\|=\rho_{k}}\Phi_{\lambda}(u)\geq \rho_{k} ^2/4>0,\quad
   \forall k\geq k_1.
\end{equation*}
\smallskip

\noindent \emph{Step 2.} We prove \eqref{eq:3.5}.
Note that for any
$k\in\{-m,-m+1,\dots,1,2,\dots\}$, $Y_{k}$ is
of finite dimension, so we can choose $M_1>0$ sufficiently large
such that
\begin{equation}\label{eq:3.13}
\|u\|\leq M_1(\int_0^{T}|u|^2)^{1/2},\quad  \forall u\in Y_{k}.
\end{equation}
By (S2) and \eqref{eq:1.3}, for the former $M_1$, there
exists a $M_2>0$ such that
\begin{equation}\label{eq:3.14}
W(t,u)\geq M_1^2|u|^2-M_2,\quad \
\forall(t,u)\in[0,T]\times\mathbb{R}^{N}.
\end{equation}
Consequently, by \eqref{eq:3.13} and \eqref{eq:3.14}, we have
\begin{equation} \label{eq:3.15}
\begin{aligned}
\Phi_{\lambda}(u)
&\leq \frac{1}{2}\|u^{+}\|^2-\frac{1}{2}\|u^{-}\|^2-\int_0^{T}W(t,u)dt \\
&\leq \frac{1}{2}\|u^{+}\|^2-\frac{1}{2}\|u^{-}\|^2-M_1^2\int_0^{T}|u|^2dt+M_2T \\
&\leq \frac{1}{2}\|u^{+}\|^2-\frac{1}{2}\|u^{-}\|^2-M_1^2
 (\frac{1}{M_1^2}\|u^{+}\|^2+\frac{1}{M_1^2}\|u^0\|^2)+M_2T  \\
&\leq -\frac{1}{2}\|u^{+}\|^2-\frac{1}{2}\|u^{-}\|^2-\|u^0\|^2+M_2T  \\
&\leq -\frac{1}{2}\|u\|^2+M_2T
\end{aligned}
\end{equation}
for all $u=u^{-}+u^0+u^{+}\in Y_{k}$. Now for any
$k\in\{-m,-m+1,\dots,1,2,\dots\}$, if we
choose
$$
r_{k}>\max\{\rho_{k},\sqrt{2M_2T}\}
$$
then \eqref{eq:3.15} implies
$$
\beta_{k}(\lambda):=\max_{u\in Y_{k},\|u\|=r_{k}}\Phi_{\lambda}(u)<0,\quad
  \forall k\in \mathbb{N}.
$$
The proof is complete.
\end{proof}

Now we prove our main result.

\begin{proof}[Proof of Theorem \ref{thm:1.1}]
 In view of \eqref{eq:1.3}, \eqref{eq:2.4} and \eqref{eq:2.6},
 $\Phi_{\lambda}$ maps bounded sets to bounded sets uniformly for 
$\lambda\in[1,2]$.
By the evenness of $W(t,u)$ in $u$, it holds that
$\Phi_{\lambda}(-u)=\Phi_{\lambda}(u)$ for all
$(\lambda,u)\in[1,2]\times E$. Therefore condition (F1) of
Theorem \ref{thm:2.1} holds. Lemma \ref{Lemma:3.1} shows that 
condition (F2) holds, whereas Lemma \ref{Lemma:3.2} implies
that condition (F3) holds for all $k\geq k_1$, where $k_1$
is given in Lemma \ref{Lemma:3.2}. Thus, by Theorem \ref{thm:2.1},
for each $k\geq k_1$ and a.e. $\lambda\in[1,2]$, there exists a
sequence $\{u_{m}^{k}(\lambda)\}_{m=1}^{\infty}\subset E$ such that
\begin{equation}\label{eq:3.16}
\sup_{m}\|u_{m}^{k}(\lambda)\|<\infty,\
\Phi'_{\lambda}(u_{m}^{k}(\lambda))\to0\ and\
\Phi_{\lambda}(u_{m}^{k}(\lambda))\to\zeta_{k}(\lambda)\quad
\end{equation}
 as $m\to\infty$, where
\begin{equation*}
\zeta_{k}(\lambda):=\inf_{\gamma\in\Gamma_{k}}\max_{u\in
B_{k}}\Phi_{\lambda}(\gamma(u)),\quad \forall\lambda\in[1,2]
\end{equation*}
with $B_{k}=\{u\in Y_{k}:\|u\|\leq r_{k}\}$ and
$\Gamma_{k}:=\{\gamma\in C(B_{k},E):\gamma\text{ is odd, }
\gamma\mid_{\partial B_{k}}=\text{id}\}$.

Moreover, by the proof of Lemma \ref{Lemma:3.2}, we have
\begin{equation}\label{eq:3.17}
\zeta_{k}(\lambda)\in[\overline{\alpha}_{k},\overline{\zeta}_{k}],\quad 
\forall k\geq k_1,
\end{equation}
where $\overline{\zeta}_{k}:=\max_{u\in B_{k}}\Phi_1(u)$ and
$\overline{\alpha}_{k}:=\rho_{k}^{^2}/4\to\infty$ as
$k\to\infty$ by \eqref{eq:3.12}.

Since the sequence $\{u_{m}^{k}(\lambda)\}_{m=1}^{\infty}$ obtained
by \eqref{eq:3.16} is bounded, it is clear that for each 
$ k\geq k_1$, we can choose $\lambda_{n}\to1$ such that the
sequence $\{u_{m}^{k}(\lambda_{n})\}_{m=1}^{\infty}$ has a strong
convergent subsequence.

In fact, without loss of generality, we assume that
\begin{equation}\label{eq:3.18}
u_{m}^{k}(\lambda_{n})^{-}\to u_0^{k}(\lambda_{n})^{-},\quad
u_{m}^{k}(\lambda_{n})^0\to u_0^{k}(\lambda_{n})^0,\quad
u_{m}^{k}(\lambda_{n})^{+}\rightharpoonup
u_0^{k}(\lambda_{n})^{+}
\end{equation}
 as $m\to\infty$ and
\begin{equation}\label{eq:3.19}
u_{m}^{k}(\lambda_{n})\rightharpoonup u_0^{k}(\lambda_{n})\quad 
\text{as }m\to\infty
\end{equation}
for some
$u_0^{k}(\lambda_{n})=u_0^{k}(\lambda_{n})^{-}
+u_0^{k}(\lambda_{n})^0+u_0^{k}(\lambda_{n})^{+}\in
E=E^{-}\oplus E^0\oplus E^{+}$ since $\dim(E^{-}\oplus E^0)<\infty$.
Note that
\begin{equation*}
\Phi'_{\lambda_{n}}(u_{m}^{k}(\lambda_{n}))
=u_{m}^{k}(\lambda_{n})^{+}-\lambda_{n}(u_{m}^{k}(\lambda_{n})^{-}
+\Psi'(u_{m}^{k}(\lambda_{n})),\quad \forall n\in \mathbb{N}.
\end{equation*}
That is,
\begin{equation}\label{eq:3.20}
u_{m}^{k}(\lambda_{n})^{+}=\Phi'_{\lambda_{n}}(u_{m}^{k}(\lambda_{n}))
+\lambda_{n}(u_{m}^{k}(\lambda_{n})^{-}+\Psi'(u_{m}^{k}(\lambda_{n})),\quad 
\forall m\in \mathbb{N}.
\end{equation}
In view of \eqref{eq:3.16}, \eqref{eq:3.18}, \eqref{eq:3.19} and the
compactness of $\Psi'$, the right-hand side of \eqref{eq:3.20}
converges strongly in $E$ and hence
$u_{m}^{k}(\lambda_{n})^{+}\to u_0^{k}(\lambda_{n})^{+}$
in $E$. Together with \eqref{eq:3.18},
$\{u_{m}^{k}(\lambda_{n})\}_{m=1}^{\infty}$ has a strong convergent
subsequence in $E$.

Without loss of generality, we may assume that
$$
\lim_{m\to\infty}u_{m}^{k}(\lambda_{n})=u_{n}^{k},\quad 
 \forall n\in\mathbb{N}\ and\ k\geq k_1.
$$
This together with \eqref{eq:3.16} and \eqref{eq:3.17} yields
\begin{equation}\label{eq:3.21}
\Phi'_{\lambda_{n}}(u_{n}^{k})=0,\ \
\Phi_{\lambda_{n}}(u_{n}^{k})\in[\overline{\alpha}_{k},\overline{\zeta}_{k}],
\quad \forall n\in\mathbb{N}\ and\ k\geq k_1.
\end{equation}
Now we claim that the sequence $\{u_{n}^{k}\}_{n=1}^{\infty}$ in
\eqref{eq:3.21} is bounded in $E$ and possesses a strong convergent
subsequence with the limit $u^{k}\in E$ for all $k\geq k_1$. For
the sake of notational simplicity, throughout the remaining proof of
Theorem \ref{thm:1.1} we  denote $u_{n}=u_{n}^{k}$. For $u_{n}\in E$,
let $\overline{u}_{n}=\frac{1}{T}\int_{o}^{T}u_{n}(t)dt$,
$u_{n}=\widetilde{u}_{n}+\overline{u}_{n}$. By \eqref{eq:2.4}, there
exists a constant $\tau_{\infty}$ for any $u\in E$ such that
\begin{equation}\label{eq:3.22}
|u|_{\infty}\leq\tau_{\infty}\|u\|.
\end{equation}

Assume by contradiction, first, we prove that $\{u_{n}\}$ is bounded
in $E$. Otherwise, going to a subsequence if necessary, we can
assume that $\|u_{n}\|\to\infty$ as $n\to\infty$.
Put $v_{n}=\frac{u_{n}}{\|u_{n}\|}$, then $v_{n}$ is bounded in $E$.
Hence, there exists a subsequence, still denoted by $v_{n}$, such
that
$$
v_{n}\rightharpoonup v_0\ \ in\ E,\ \ v_{n}\to v_0\ \
in\ C([0,T],\mathbb{R}^{N}).
$$
Then, we have
\begin{equation}\label{eq:3.23}
\overline{v}_{n}\to \overline{v}_0.
\end{equation}

By \eqref{eq:1.3}, for all $|u|\leq L$, we have
\begin{equation*}
W(t,u)\leq d_1(|u|+|u|^{\alpha+1})+d_2\leq
d_1(L+L^{\alpha+1})+d_2,
\end{equation*}
which together with (S3) yields
\begin{equation}\label{eq:3.24}
\mu W(t,u)\leq \langle\nabla_{u}W(t,u),u\rangle+a(t)|u|^{\beta}+\mu
d_1(L+L^{\alpha+1})+\mu d_2,
\end{equation}
for all $u\in\mathbb{R}^{N}$ and $t\in[0,T]$. It follows from
\eqref{eq:2.6}, \eqref{eq:2.7} that
\begin{align*}
\mu\Phi_{\lambda_{n}}(u_{n})-\langle\Phi'_{\lambda_{n}}(u_{n}),u_{n}\rangle
&=(\frac{\mu}{2}-1)(\|u_{n}^{+}\|^2-\|u_{n}^{-}\|^2)
 -(\lambda_{n}-1)(\frac{\mu}{2}-1)\|u_{n}^{-}\|^2\\
&\quad -\lambda_{n}\int_0^{T}(\mu
W(t,u_{n})-\langle\nabla_{u}W(t,u_{n}),u_{n}\rangle)dt\,.
\end{align*}
In the following, we denote $C_{i}>0(i=0,1,2,\dots)$ for different
positive constants. Comparing \eqref{eq:2.1} with \eqref{eq:2.2}, we
learn that
\begin{align*}
&(\frac{\mu}{2}-1)\|\dot{u_{n}}\|_{L^2}^2\\
&=\mu\Phi_{\lambda_{n}}(u_{n})-\langle\Phi'_{\lambda_{n}}(u_{n}),u_{n}\rangle
-(\frac{\mu}{2}-1)\int_0^{T}\langle U(t)u_{n},u_{n}\rangle
dt\\
&\quad +(\lambda_{n}-1)(\frac{\mu}{2}-1)\|u_{n}^{-}\|^2+\lambda_{n}\int_0^{T}(\mu
W(t,u_{n})-\langle\nabla_{u}W(t,u_{n}),u_{n}\rangle)dt.
\end{align*}
This together with the positive definite assumption of matrix $U$,
\eqref{eq:2.4}, \eqref{eq:3.21}, \eqref{eq:3.24} and $\mu>2$ implies
\begin{equation} \label{eq:3.25}
\begin{aligned}
(\frac{\mu}{2}-1)\|\dot{u_{n}}\|_{L^2}^2
&\leq C_1+(\frac{\mu}{2}-1)(\lambda_{n}-1)\|u_{n}^{-}\|^2\\
&\quad +\lambda_{n}\int_0^{T}(a(t)|u_{n}|^{\beta}+\mu
d_1(L+L^{\alpha+1})+\mu d_2)dt\\
&\leq C_2+C_3(\lambda_{n}-1)\|u_{n}^{-}\|^2+C_{4}\|u_{n}\|^{\beta}.
\end{aligned}
\end{equation}
Note that $0<\beta<2$, $\lambda_{n}\to1$ and
$\|u_{n}^{-}\|^2\leq\|u_{n}\|^2$ , we have
\begin{equation*}
\frac{\|\dot{u_{n}}\|_{L^2}^2}{\|u_{n}\|^2}\leq
\frac{C_{5}}{\|u_{n}\|^2}+
C_{6}(\lambda_{n}-1)\frac{\|u_{n}^{-}\|^2}{\|u_{n}\|^2}
+C_{7}\frac{\|u_{n}\|^{\beta}}{\|u_{n}\|^2}\to 0\quad \text{as }
n\to\infty;
\end{equation*}
i.e., $\|\dot{v_{n}}\|_{L^2}\to 0$ as $n\to\infty$.
Together with \eqref{eq:3.23}, we have
$v_{n}\to \overline{v}_0$  as $n\to\infty$.
Therefore, we obtain
$$
v_0=\overline{v}_0,\quad  T|\overline{v}_0|^2=\|\overline{v}_0\|^2=1\,.
$$
Consequently, $|u_{n}|\to\infty$ as $n\to\infty$
uniformly for a.e. $t\in[0.T]$. From (S2), we obtain
\begin{align*}
\liminf_{|u_{n}|\to\infty}\frac
{\int_0^{T}W(t,u_{n})dt}{\|u_{n}\|^2}
&\geq \frac {\int_0^{T}\liminf_{|u_{n}|\to\infty}
W(t,u_{n})dt}{\|u_{n}\|^2}\\
&=\int_0^{T}[\liminf_{|u_{n}|\to\infty}\frac
{W(t,u_{n})}{|u_{n}|^2}|v_{n}|^2]dt\\
&= \int_0^{T}[\liminf_{|u_{n}|\to\infty}\frac
{W(t,u_{n})}{|u_{n}|^2}|v_0|^2]dt>0\,.
\end{align*}
Hence,
\begin{equation} \label{eq:3.26}
\liminf_{|u_{n}|\to\infty}\frac{\int_0^{T}W(t,u_{n})dt}{\|u_{n}\|^2}
>0.
\end{equation}
On the other hand, from \eqref{eq:2.4}, \eqref{eq:2.6},
\eqref{eq:2.7}, \eqref{eq:3.21}and \eqref{eq:3.24}, we have
\begin{align*}
(\frac{\mu}{2}-1)(\|u^{+}_{n}\|^2-\lambda_{n}\|u^{-}_{n}\|^2)
&=\mu\Phi_{\lambda_{n}}(u_{n})-\langle\Phi'_{\lambda_{n}}(u_{n}),u_{n}\rangle\\
&\quad +\lambda_{n}\int_0^{T}(\mu
W(t,u_{n})-\langle\nabla_{u}W(t,u_{n}),u_{n}\rangle)dt\\
&\leq C_1+\lambda_{n}\int_0^{T}(a(t)|u_{n}|^{\beta}+\mu
d_1(L+L^{\alpha+1})+\mu d_2)dt\\
&\leq C_{8}+C_{9}\lambda_{n}\|u_{n}\|^{\beta}\,.
\end{align*}
Note that $\mu>2$; then we obtain
\begin{equation}\label{eq:3.27}
(\|u^{+}_{n}\|^2-\lambda_{n}\|u^{-}_{n}\|^2)
\leq\frac{2C_{8}}{\mu-2}+\frac{2C_{9}}{\mu-2}\lambda_{n}\|u_{n}\|^{\beta}.
\end{equation}
By the boundedness of $\Phi_{\lambda_{n}}(u_{n})$, and
\eqref{eq:3.27}, we have
\begin{align*}
\frac{\Phi_{\lambda_{n}}(u_{n})}{\|u_{n}\|^2}
&=\frac{\frac{1}{2}(\|u^{+}_{n}\|^2-\lambda_{n}\|u^{-}_{n}\|^2)}{\|u_{n}\|^2}
  -\frac{\lambda_{n}\int_0^{T}W(t,u_{n})dt}{\|u_{n}\|^2}\\
&\leq \frac{\frac{C_{8}}{\mu-2}}{\|u_{n}\|^2}+\frac{\frac{C_{9}}{\mu-2}\lambda_{n}\|u_{n}\|^{\beta}}{\|u_{n}\|^2}
  -\frac{\lambda_{n}\int_0^{T}W(t,u_{n})dt}{\|u_{n}\|^2},
\end{align*}
which together with $0<\beta<2$ implies
$$
\lim_{|u_{n}|\to\infty}\inf\frac{\int_0^{T}W(t,u_{n})dt}{\|u_{n}\|^2}=0.
$$
This contradicts to \eqref{eq:3.26}. Thus, $\{u_{n}\}$ is bounded in
$E$.

The proof that $\{u_{n}\}$ has a strong convergent subsequence
is the same as the preceding proof of
$\{u_{m}^{k}(\lambda_{n})\}_{m=1}^{\infty}$.

Now for each $k\geq k_1$, by \eqref{eq:3.21}, the limit $u^{k}$ is
just a critical point of $\Phi=\Phi_1$ with
$\Phi(u^{k})\in[\overline{\alpha}_{k},\overline{\zeta}_{k}]$. Since
$\overline{\alpha}_{k}\to\infty$ as $k\to\infty$ in
\eqref{eq:3.17}, we obtain infinitely many nontrivial critical points
of $\Phi$. Therefore, system \eqref{eq:1.1} possesses infinitely
many nontrivial solutions. The proof of Theorem \ref{thm:1.1} is
complete.
\end{proof}

\subsection*{Acknowledgements}
The authors sincerely thank the referee for his/her careful reading
and helpful and insightful comments of the manuscript. The work is
supported by the Fundamental Research Funds for the Central
Universities and the National Natural Science Foundation of China
(No. 61001139).


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\end{document}