\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 $$\label{eq:1.1} \begin{gathered} \ddot{u}(t)-U(t)u(t)+\nabla_{u}W(t,u)=0,\quad \forall t\in \mathbb{R},\\ u(0)=u(T),\quad \dot{u}(0)=\dot{u}(T),\quad T>0, \end{gathered}$$ 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 $$\label{eq:1.2} \begin{gathered} \ddot{u}(t)+\nabla_{u}V(t,u)=0,\quad \forall t\in \mathbb{R},\\ u(0)=u(T), \quad \dot{u}(0)=\dot{u}(T),\quad T>0, \end{gathered}$$ 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 $$\label{eq:1.3} |W(t,u)|\leq d_1(|u|+|u|^{\alpha+1})+d_2,\quad \forall(t,u)\in[0,T]\times\mathbb{R}^{N}.$$ 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 $$\label{eq:1.4} 0<\mu W(t,u)\leq \langle\nabla_{u}W(t,u),u\rangle,\quad \forall (t,u)\in\mathbb{R}\times\mathbb{R}^{N}\setminus\{0\}.$$ 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 $$\label{eq:2.1} \Phi(u)=\frac{1}{2}\Big(\int_0^{T}|\dot{u}|^2dt+\int_0^{T}\langle U(t)u,u\rangle dt\Big)-\Psi(u),$$ 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 $$\label{eq:2.2} \Phi(u)=\frac{1}{2}(\|u^{+}\|^2-\|u^{-}\|^2)-\Psi(u).$$ Direct computations show that $$\label{eq:2.3} \begin{gathered} \langle\Psi'(u), v\rangle = \int_0^{T}\langle\nabla_{u}W(t,u),v\rangle dt \\ \langle\Phi'(u), v\rangle = \langle u^{+},v^{+}\rangle_0-\langle u^{-},v^{-}\rangle_0-\langle\Psi'(u), v\rangle \end{gathered}$$ 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 $$\label{eq:2.4} |u|_{p}\leq\tau_{p}\|u\|,\quad \forall u\in E.$$\ 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 $$\label{eq:2.7} \langle\Phi'_{\lambda}(u), v\rangle=\langle u^{+},v^{+}\rangle_0-\lambda(\langle u^{-},v^{-}\rangle_0+\int_0^{T}\langle\nabla_{u}W(t,u),v\rangle dt).$$ 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 $$\label{eq:3.1} m(\{t\in[0,T]:|u|\geq\epsilon\|u\|\})\geq\epsilon,\quad \forall u\in F\setminus\{0\},$$ 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 $$\label{eq:3.2} W(t,u)\geq d_3|u|^2, \quad \ \forall t\in[0,T]\ and\ |u|\geq R_1.$$ Note that $$\label{eq:3.3} |u(t)|\geq R_1, \quad \ \forall t\in\Lambda_{u}$$ 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 $$\label{eq:3.4} \alpha_{k}(\lambda):=\inf_{u\in Z_{k},\|u\|=\rho_{k}}\Phi_{\lambda}(u)>0,\quad \forall k\geq k_1,$$ and $$\label{eq:3.5} \beta_{k}(\lambda):=\max_{u\in Y_{k},\|u\|=r_{k}}\Phi_{\lambda}(u)<0,\quad \forall k\in \mathbb{N},$$ 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 \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} where $d_1$, $d_2$ are the constants in \eqref{eq:1.3}. Let $$\label{eq:3.7} \iota_{\alpha+1}(k)=\sup_{u\in Z_{k},\|u\|=1}|u|_{\alpha+1},\quad \forall k\in \mathbb{N}.$$ Then $$\label{eq:3.8} \iota_{\alpha+1}(k)\to0\quad \text{as } k\to\infty$$ since $E$ is compactly embedded into $L^{\alpha+1}$. Note that $$\label{eq:3.9} Z_{k}\subset E^{+},\quad \forall k\geq1.$$ Combining \eqref{eq:2.4}, \eqref{eq:3.6}, \eqref{eq:3.7} and \eqref{eq:3.9}, for $k\geq1$, we have $$\label{eq:3.10} \Phi_{\lambda}(u)\geq\frac{1}{2}\|u\|^2-2d_1\tau_1\|u\| -2d_2T-2d_1\iota_{\alpha+1}^{\alpha+1}(k)\|u\|^{\alpha+1},$$ 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 $$\label{eq:3.11} \rho_{k}:=(16d_1\iota_{\alpha+1}^{\alpha+1}(k))^{1/(1-\alpha)} >\max\{16d_1\tau_1+1,16d_2T\},\quad \forall k\geq k_1$$ since $\alpha>1$. Clearly, $$\label{eq:3.12} \rho_{k}\to\infty\quad \ as\ k\to\infty.$$ 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 $$\label{eq:3.13} \|u\|\leq M_1(\int_0^{T}|u|^2)^{1/2},\quad \forall u\in Y_{k}.$$ By (S2) and \eqref{eq:1.3}, for the former $M_1$, there exists a $M_2>0$ such that $$\label{eq:3.14} W(t,u)\geq M_1^2|u|^2-M_2,\quad \ \forall(t,u)\in[0,T]\times\mathbb{R}^{N}.$$ Consequently, by \eqref{eq:3.13} and \eqref{eq:3.14}, we have \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} 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 $$\label{eq:3.16} \sup_{m}\|u_{m}^{k}(\lambda)\|<\infty,\ \Phi'_{\lambda}(u_{m}^{k}(\lambda))\to0\ and\ \Phi_{\lambda}(u_{m}^{k}(\lambda))\to\zeta_{k}(\lambda)\quad$$ 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 $$\label{eq:3.17} \zeta_{k}(\lambda)\in[\overline{\alpha}_{k},\overline{\zeta}_{k}],\quad \forall k\geq k_1,$$ 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 $$\label{eq:3.18} u_{m}^{k}(\lambda_{n})^{-}\to u_0^{k}(\lambda_{n})^{-},\quad u_{m}^{k}(\lambda_{n})^0\to u_0^{k}(\lambda_{n})^0,\quad u_{m}^{k}(\lambda_{n})^{+}\rightharpoonup u_0^{k}(\lambda_{n})^{+}$$ as $m\to\infty$ and $$\label{eq:3.19} u_{m}^{k}(\lambda_{n})\rightharpoonup u_0^{k}(\lambda_{n})\quad \text{as }m\to\infty$$ 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, $$\label{eq:3.20} u_{m}^{k}(\lambda_{n})^{+}=\Phi'_{\lambda_{n}}(u_{m}^{k}(\lambda_{n})) +\lambda_{n}(u_{m}^{k}(\lambda_{n})^{-}+\Psi'(u_{m}^{k}(\lambda_{n})),\quad \forall m\in \mathbb{N}.$$ 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 $$\label{eq:3.21} \Phi'_{\lambda_{n}}(u_{n}^{k})=0,\ \ \Phi_{\lambda_{n}}(u_{n}^{k})\in[\overline{\alpha}_{k},\overline{\zeta}_{k}], \quad \forall n\in\mathbb{N}\ and\ k\geq k_1.$$ 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 $$\label{eq:3.22} |u|_{\infty}\leq\tau_{\infty}\|u\|.$$ 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 $$\label{eq:3.23} \overline{v}_{n}\to \overline{v}_0.$$ 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 $$\label{eq:3.24} \mu W(t,u)\leq \langle\nabla_{u}W(t,u),u\rangle+a(t)|u|^{\beta}+\mu d_1(L+L^{\alpha+1})+\mu d_2,$$ 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 \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} 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, $$\label{eq:3.26} \liminf_{|u_{n}|\to\infty}\frac{\int_0^{T}W(t,u_{n})dt}{\|u_{n}\|^2} >0.$$ 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 $$\label{eq:3.27} (\|u^{+}_{n}\|^2-\lambda_{n}\|u^{-}_{n}\|^2) \leq\frac{2C_{8}}{\mu-2}+\frac{2C_{9}}{\mu-2}\lambda_{n}\|u_{n}\|^{\beta}.$$ 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). \begin{thebibliography}{10} \bibitem{1} A. 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