\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 128, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/128\hfil Homoclinic solutions] {Homoclinic solutions for a class of second order non-autonomous systems} \author[R. Yuan, Z. Zhang\hfil EJDE-2009/128\hfilneg] {Rong Yuan, Ziheng Zhang} % in alphabetical order \address{Rong Yuan \newline School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China} \email{ryuan@bnu.edu.cn} \address{Ziheng Zhang \newline School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China} \email{zhzh@mail.bnu.edu.cn} \thanks{Submitted October 14, 2008. Published October 7, 2009.} \thanks{Supported by National Natural Science Foundation of China and RFDP} \subjclass[2000]{34C37, 35A15, 37J45} \keywords{Homoclinic solutions; critical point; variational methods; \hfill\break\indent mountain pass theorem} \begin{abstract} This article concerns the existence of homoclinic solutions for the second order non-autonomous system $$\ddot q+A \dot q-L(t)q+W_{q}(t,q)=0,$$ where $A$ is a skew-symmetric constant matrix, $L(t)$ is a symmetric positive definite matrix depending continuously on $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$. We assume that $W(t,q)$ satisfies the global Ambrosetti-Rabinowitz condition, that the norm of $A$ is sufficiently small and that $L$ and $W$ satisfy additional hypotheses. We prove the existence of at least one nontrivial homoclinic solution, and the existence of infinitely many homoclinic solutions if $W(t,q)$ is even in $q$. Recent results in the literature are generalized and improved. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} The purpose of this work is to study the existence of {\it homoclinic } solutions for the second order non-autonomous system $$\ddot q+A \dot q-L(t)q+W_{q}(t,q)=0,\label{DS}$$ where $A$ is a skew-symmetric constant matrix, $L(t)$ is a symmetric and positive definite matrix depending continuously on $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$. A solution $q(t)$ of \eqref{DS} is called a homoclinic solution (to 0) if $q\in C^2(\mathbb{R},\mathbb{R}^n)$, $q(t)\to 0$ and $\dot q(t)\to 0$ as $t\to\pm\infty$. If $q(t)\not\equiv 0$, $q(t)$ is called a nontrivial homoclinic solution. When $A=0$, \eqref{DS} is the second order Hamiltonian system. Assuming that $L(t)$ and $W(t,q)$ are independent of $t$ or $T$-periodic in $t$, the existence of homoclinic solutions for the Hamiltonian system \eqref{DS} has been studied via critical point theory and variational methods, see for instance \cite{Am,Caldiroli,Co2,Di2,Fl,Paturel,Rab3} and the references therein; a more general case is considered in \cite{Iz}. In this case, the existence of homoclinic solutions can be obtained by taking the limit of periodic solutions of approximating problems. If $L(t)$ and $W(t,q)$ are neither independent of $t$ not periodic in $t$, compactness arguments derived from Sobolev imbedding theorem are not available for the study of \eqref{DS}, see \cite{Al,Ca,Di1,Korman,Lv,Om,Ou,Rab1} and the references therein. When $A\neq 0$, as far as we know, the existence of homoclinic solutions of \eqref{DS} has not been studied. Our basic hypotheses on $L$ and $W$ are: \begin{itemize} \item [(H1)] $L\in C(\mathbb{R},\mathbb{R}^{n^2})$, $L(t)$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, and there is a continuous function $\alpha:\mathbb{R}\to \mathbb{R}$ such that $\alpha(t)>0$ for all $t\in \mathbb{R}$, $\big(L(t)q,q\big)\geq \alpha(t)|q|^2$, and $\alpha(t)\to +\infty$ as $|t|\to +\infty$. \item[(H2)]There exists a constant $\mu>2$ such that for every $t\in \mathbb{R}$ and $q\in \mathbb{R}^n\backslash\{0\}$ $$0<\mu W(t,q)\leq\big(W_{q}(t,q),q\big).$$ \item[(H3)] $W_q(t,q)=o(|q|)$ as $|q|\to 0$ uniformly with respect to $t\in \mathbb{R}$. \item[(H4)]There exists $\overline{W}\in C(\mathbb{R}^n,\mathbb{R})$ such that $|W_q(t,q)|\leq | \overline{W}(q)|$ for every $t\in \mathbb{R}$ and $q\in \mathbb{R}^n$. \end{itemize} \begin{remark}\label{rem 1.1} {\rm From (H1), we see that there is a constant $\beta>0$ such that $$\label{1.1} \big(L(t)q,q\big)\geq \beta|q|^2 \quad \text{for all t\in \mathbb{R} and q\in \mathbb{R}^n}.$$ (H2) is called the global Ambrosetti-Rabinowitz condition due to Ambrosetti and Rabinowitz (e.g., \cite{AmR}). Combining (H2) with (H3), we see that $W(t,q)\geq 0$ for all $(t,q)\in \mathbb{R}\times \mathbb{R}^{n}$, $W(t,0)=0$, $W_q(t,0)=0$. Moreover, $W(t,q)=o(|q|^2)$ as $|q|\to 0$ uniformly with respect to $t$, which implies that for any $\varepsilon>0$ there is $\delta>0$ such that $$\label{1.2} W(t,q)\leq \varepsilon |q|^2\quad \text{for } (t,q)\in \mathbb{R}\times \mathbb{R}^n,\; |q|\leq \delta.$$ }\end{remark} In addition, we need the following hypothesis on $A$. \begin{itemize} \item[(H5)] $\|A\|<\sqrt{\beta}$, where $\beta$ is defined in \eqref{1.1}. \end{itemize} Now we state our main result. \begin{theorem}\label{Thm 1.1} Assume {\rm (H1)--(H5)}. Then \eqref{DS} possesses at least one nontrivial homoclinic solution. Moreover, if we assume that $W(t,q)$ is even in $q$; i.e., \begin{itemize} \item[(H6)] $W(t,-q)=W(t,q)$ for all $t\in \mathbb{R}$ and $q\in \mathbb{R}^n$, \end{itemize} then \eqref{DS} has infinitely many distinct homoclinic solutions. \end{theorem} \begin{remark} {\rm From Remark \ref{rem 1.1}, we know that there exists $\beta>0$ such that \eqref{1.1} holds. However, since we do not have an explicit estimate on $\beta$, we simply assume that $\|A\|$ is sufficiently small. Furthermore, when $A=0$, our main result is just \cite[Theorems 1 and 2]{Om}. }\end{remark} To overcome the lack of compactness in standard Sobolev imbedding theorems, we employ a compact imbedding theorem obtained in \cite{Om}. In Section 2 we state and prove preliminary results. Section 3 is devoted to the proof of Theorem \ref{Thm 1.1}. \section{Preliminaries} Let $$E=\big\{q\in H^1(\mathbb{R},\mathbb{R}^n):\int_{\mathbb{R}}\big[|\dot q(t)|^2+\big(L(t)q(t),q(t)\big) \big]dt<+\infty\big\}.$$ This vector space is a Hilbert space when endowed with the inner product $$(x,y)=\int_{\mathbb{R}}\big[\big(\dot x(t),\dot y(t)\big)+\big(L(t)x(t),y(t)\big)\big]dt$$ and the corresponding norm $\|x\|^2=(x,x)$. Note that $$E\subset H^1(\mathbb{R},\mathbb{R}^n)\subset L^p(\mathbb{R},\mathbb{R}^n)$$ for all $p\in[2,+\infty]$ with the imbedding being continuous. In particular, for $p=+\infty$, there exists a constant $C>0$ such that $$\label{2.1} \|q\|_{\infty}\leq C\|q\|, \quad \forall q\in E.$$ Here $L^p(\mathbb{R},\mathbb{R}^n)$ ($2\leq p <+\infty$) and $H^1(\mathbb{R},\mathbb{R}^n)$ denote the Banach spaces of functions on $\mathbb{R}$ with values in $\mathbb{R}^n$ under the norms $$\|q\|_p:=\big(\int_{\mathbb{R}}|q(t)|^p dt\big)^{1/p}\quad \text{and} \quad \|q\|_{H^1}:= \big(\|q\|_2^2+\|\dot q\|_2^2\big)^{1/2}$$ respectively. $L^{\infty}(\mathbb{R},\mathbb{R}^n)$ is the Banach space of essentially bounded functions from $\mathbb{R}$ into $\mathbb{R}^n$ equipped with the norm $$\|q\|_{\infty}:=\text{ess} \sup\{|q(t)|: t\in \mathbb{R} \}.$$ \begin{lemma}[{\cite[Lemma 1]{Om}}] \label{lem 2.1} Assume $L$ satisfies {\rm (H1)}. Then the embedding of $E$ in $L^2(\mathbb{R},\mathbb{R}^n)$ is compact. \end{lemma} \begin{lemma}[{\cite[Lemma 2]{Om}}] \label{lem 2.2} Assume {\rm (H1), (H3), (H4)}. If $q_k\rightharpoonup q_0$ (weakly) in $E$, then $W_q(t,q_k)\to W_q(t,q_0)$ in $L^2(\mathbb{R},\mathbb{R}^n)$. \end{lemma} \begin{lemma}\label{lem 2.3} Under Assumption {\rm (H2)}, for every $t\in\mathbb{R}$, we have \begin{gather} W(t,q)\leq W\Big(t,\frac{q}{|q|}\Big)|q|^{\mu},\quad\text{if } 0<|q|\leq 1, \\ W(t,q)\geq W\Big(t,\frac{q}{|q|}\Big)|q|^{\mu},\quad\text{if } |q|\geq 1. \end{gather} \end{lemma} \begin{proof} It suffices to show that for every $q\neq 0$ and $t\in \mathbb{R}$ the function $(0,\infty)\ni \zeta\to W(t,\zeta^{-1}q)\zeta^{\mu}$ is non-increasing, which is an immediate consequence of (H2). \end{proof} \begin{remark}\label{rem 2.1} {\rm From Lemma \ref{lem 2.3}, we see that there exists $\alpha_0(t)>0$ such that $$W(t,q)\geq \alpha_0(t)|q|^{\mu}\quad \text{for all } (t,q)\in \mathbb{R}\times \mathbb{R}^{n},\; |q|\geq 1.$$ }\end{remark} Now we introduce more notation and some definitions. Let $\mathcal{B}$ be a real Banach space, $I\in C^1(\mathcal{B},\mathbb{R})$, which means that $I$ is a continuously Fr\'echet-differentiable functional defined on $\mathcal{B}$. \begin{definition}[\cite{Rab2}] {\rm $I\in C^1(\mathcal{B},\mathbb{R})$ is said to satisfy the (PS) condition if any sequence $\{u_{j}\}_{j\in \mathbb{N}}\subset \mathcal{B}$, for which $\{I(u_{j})\}_{j\in \mathbb{N}}$ is bounded and $I'(u_{j})\to 0$ as $j\to +\infty$, possesses a convergent subsequence in $\mathcal{B}$. }\end{definition} Moreover, let $B_{r}$ be the open ball in $\mathcal{B}$ with the radius $r$ and centered at $0$ and $\partial B_{r}$ denote its boundary. We obtain the existence and multiplicity of homoclinic solutions of \eqref{DS} by use of the following well-known Mountain Pass Theorems, see \cite{Rab2}. \begin{lemma}[{\cite[Theorem 2.2]{Rab2}}] \label{lem 2.4} Let $\mathcal{B}$ be a real Banach space and $I\in C^1(\mathcal{B},\mathbb{R})$ satisfying the (PS) condition. Suppose that $I(0)=0$ and \begin{itemize} \item [(A1)] there exist constants $\rho$, $\alpha>0$ such that $I|_{\partial B_{\rho}}\geq \alpha$, \item [(A2)] there exists $e\in {\mathcal{B}}\setminus \overline{B}_{\rho}$ such that $I(e)\leq 0$. \end{itemize} Then $I$ possesses a critical value $c\geq \alpha$ given by $$c=\inf_{g\in\Gamma}\max_{s\in[0,1]}I(g(s)),$$ where $$\Gamma=\{g\in C([0,1], {\mathcal{B}}): g(0)=0,g(1)=e\}.$$ \end{lemma} \begin{lemma}[{\cite[Theorem 9.12]{Rab2}}] \label{lem 2.6} Let $\mathcal{B}$ be an infinite dimensional real Banach space and $I\in C^1(\mathcal{B},\mathbb{R})$ be even satisfying the (PS) condition and $I(0)=0$. If ${\mathcal{B}}=V\oplus X$, where $V$ is finite dimensional, and $I$ satisfies \begin{itemize} \item [(A3)] there exist constants $\rho$, $\alpha>0$ such that $I|_{\partial B_{\rho}\cap X}\geq \alpha$ and \item [(A4)] for each finite dimensional subspace $\tilde{\mathcal{B}}\subset \mathcal{B}$, there is an $R=R(\tilde{\mathcal{B}})$ such that $I\leq 0$ on $\tilde{\mathcal{B}}\backslash B_{R(\tilde{\mathcal{B}})}$, \end{itemize} then $I$ has an unbounded sequence of critical values. \end{lemma} \section{Proof of Theorem \ref{Thm 1.1}} Now we establish the corresponding variational framework to obtain homoclinic solutions of \eqref{DS}. Take ${\mathcal{B}}=E$ and define the functional $I:E\to \mathbb{R}$ by \label{3.1} \begin{aligned} I(q)&= \int_{\mathbb{R}}\Big[\frac{1}{2}|\dot q(t)|^2 +\frac{1}{2}\big(A q(t),\dot q(t)\big) +\frac{1}{2}\big(L(t)q(t),q(t)\big)-W(t,q(t))\Big]dt\\ &= \frac{1}{2}\|q\|^2+\frac{1}{2}\int_{\mathbb{R}}\big(A q(t),\dot q(t)\big)dt-\int_{\mathbb{R}}W(t,q(t))dt. \end{aligned} \begin{lemma}\label{lem 3.0} Under the conditions of Theorem \ref{Thm 1.1}, we have $$\label{3.2} I'(q)v=\int_{\mathbb{R}}\Big[\big(\dot q(t),\dot v(t)\big)-\big(A \dot q(t), v(t)\big)+\big(L(t)q(t),v(t)\big)-\Big(W_{q}(t,q(t)),v(t)\Big)\Big]dt,$$ for all $q$, $v\in E$, which yields, using the skew-symmetry of $A$, \label{3.3} \begin{aligned} I'(q)q &= \|q\|^2-\int_{\mathbb{R}}\big(A\dot q(t),q(t)\big)dt -\int_{\mathbb{R}}\Big(W_{q}(t,q(t)),q(t)\Big)dt\\ &= \|q\|^2+\int_{\mathbb{R}}\big(A q(t),\dot q(t)\big)dt -\int_{\mathbb{R}}\Big(W_{q}(t,q(t)),q(t)\Big)dt. \end{aligned} Moreover, $I$ is a continuously Fr\'echet-differentiable functional defined on $E$; i.e., $I\in C^1(E,\mathbb{R})$ and any critical point of $I$ on $E$ is a classical solution of \eqref{DS} with $q(\pm\infty)=0=\dot q(\pm\infty)$. \end{lemma} \begin{proof} We begin by showing that $I:E\to \mathbb{R}$. By (\ref{1.2}), there exist constants $M>0$ and $R_1>0$ such that $$\label{3.4} W(t,q)\leq M |q|^2 \quad \text{for all }(t,q)\in \mathbb{R}\times \mathbb{R}^{n},\; |q|\leq R_1.$$ Letting $q\in E$, then $q\in C^0(\mathbb{R},\mathbb{R}^n)$ (see, e.g., \cite{Rab3}), the space of continuous functions $q$ on $\mathbb{R}$ such that $q(t)\to 0$ as $|t|\to +\infty$; i.e., $E\subset C^0(\mathbb{R},\mathbb{R}^n)$. Therefore there is a constant $R_2> 0$ such that $|t|\geq R_2$ implies that $|q(t)|\leq R_1$. Hence, by (\ref{3.4}), we have $$\label{3.5} 0\leq\int_{\mathbb{R}}W(t,q(t))dt\leq\int_{-R_2}^{R_2}W(t,q(t))dt+M \int_{|t|\geq R_2}|q(t)|^2dt<+\infty.$$ Combining (\ref{3.1}) and (\ref{3.5}), we show that $I:E\to \mathbb{R}$. Next we prove that $I\in C^1(E,\mathbb{R})$. Rewrite $I$ as $I=I_1-I_2$, where $$I_1:=\frac{1}{2}\int_{\mathbb{R}}\Big[|\dot q(t)|^2+\big(Aq(t),\dot q(t)\big) +\big(L(t)q(t),q(t)\big)\Big]dt,\quad I_2:=\int_{\mathbb{R}}W(t,q(t))dt.$$ It is easy to check that $I_1\in C^1(E,\mathbb{R})$, and by using the skew-symmetry of $A$, we have $$\label{3.6} I_1'(q)v=\int_{\mathbb{R}}\Big[\big(\dot q(t),\dot v(t)\big)-\big(A\dot q(t), v(t)\big)+\big(L(t)q(t),v(t)\big)\Big]dt.$$ Therefore it is sufficient to consider $I_2$. In the process we will see that $$\label{3.7} I_2'(q)v=\int_{\mathbb{R}}\big(W_q(t,q(t)), v(t)\big)dt,$$ which is defined for all $q$, $v\in E$. For any given $q\in E$, let us define $J(q):E\to \mathbb{R}$ as following $$J(q)v=\int_{\mathbb{R}}\big(W_q(t,q(t)), v(t)\big)dt, \quad v\in E.$$ It is obvious that $J(q)$ is linear. Now we show that $J(q)$ is bounded. Indeed, for any given $q\in E$, there exists a constant $M_1>0$ such that $\|q\|\leq M_1$ and, by (\ref{2.1}), $\|q\|_{\infty}\leq C M_1$. According to (H3) and (H4), there is a constant $b_1>0$ (dependent on $q$) such that $$|W_q(t,q(t))|\leq b_1|q(t)| \quad \text{for all } t\in \mathbb{R},$$ which by \eqref{1.1} and the H\"{o}lder inequality yields $$\label{3.8} |J(q)v|=\Big|\int_{\mathbb{R}}\big(W_q(t,q(t)), v(t)\big)dt\Big| \leq b_1 \|q\|_2 \,\|v\|_2 \leq \frac{b_1}{\beta}\|q\|\,\|v\|.$$ Moreover, for $q$ and $v\in E$, by the Mean Value Theorem, we have $$\int_{\mathbb{R}}W(t,q(t)+v(t))dt-\int_{\mathbb{R}}W(t,q(t))dt =\int_{\mathbb{R}}\big(W_q(t,q(t)+h(t)v(t)), v(t)\big)dt,$$ where $h(t)\in (0,1)$. Therefore, by Lemma \ref{lem 2.2} and the H\"{o}lder inequality, we have \label{3.9} \begin{aligned} &\int_{\mathbb{R}}\big(W_q(t,q(t)+h(t)v(t)), v(t)\big)dt -\int_{\mathbb{R}}\big(W_q(t,q(t)), v(t)\big)dt\\ &=\int_{\mathbb{R}}\big(W_q(t,q(t)+h(t)v(t))-W_q(t,q(t)),v(t)\big) dt\to 0 \end{aligned} as $v\to 0$. Combining (\ref{3.8}) and (\ref{3.9}), we see that (\ref{3.7}) holds. It remains to prove that $I_2'$ is continuous. Suppose that $q\to q_0$ in $E$ and note that $$I_2'(q)v-I_2'(q_0)v=\int_{\mathbb{R}}\big(W_q(t,q(t))-W_q(t,q_0(t)), v(t)\big)dt.$$ By Lemma \ref{lem 2.2} and the H\"{o}lder inequality, we obtain $$I_2'(q)v-I_2'(q_0)v\to0 \quad \text{as } q\to q_0,$$ which implies the continuity of $I_2'$ and we show that $I\in C^1(E,\mathbb{R})$. Lastly, we check that critical points of $I$ are classical solutions of \eqref{DS} satisfying $q(t)\to 0$ and $\dot q(t)\to 0$ as $|t|\to +\infty$. It is well known that $E\subset C^0(\mathbb{R},\mathbb{R}^n)$ (the space of continuous functions $q$ on $\mathbb{R}$ such that $q(t)\to 0$ as $|t|\to +\infty$). On the other hand, if $q$ is a critical point of $I$, for any $v\in E\subset C^0(\mathbb{R},\mathbb{R}^n)$, by (\ref{3.2}) we have \begin{align*} \int_{\mathbb{R}}\Big[\big(\dot q(t),\dot v(t)\big)-\big(A \dot q(t), v(t)\big)\Big]dt &= \int_{\mathbb{R}}\big(\dot q(t)+Aq(t),\dot v(t)\big)dt\\ &= -\int_{\mathbb{R}}\big(L(t)q(t)-W_{q}(t,q(t)),v(t)\big)dt, \end{align*} which implies that $L(t) q-W_{q}(t,q)$ is the weak derivative of $\dot q +A q$. Since $L\in C(\mathbb{R},\mathbb{R}^{n^2})$, $W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})$ and $E\subset C^0(\mathbb{R},\mathbb{R}^n)$, we see that $\dot q +A q$ is continuous, which yields that $\dot q$ is continuous and $q\in C^2(\mathbb{R},\mathbb{R}^n)$; i.e., $q$ is a classical solution of \eqref{DS}. Moreover, it is easy to check that $q$ satisfies $\dot q(t)\to 0$ as $|t|\to +\infty$ since $\dot q$ is continuous. \end{proof} \begin{lemma}\label{lem 3.1} Under Assumption {\rm (H1)-(H5)}, $I$ satisfies the (PS) condition. \end{lemma} \begin{proof} Assume that $\{u_{j}\}_{j\in \mathbb{N}} \subset E$ is a sequence such that $\{I(u_{j})\}_{j\in \mathbb{N}}$ is bounded and $I'(u_{j})\to 0$ as $j\to +\infty$. Then there exists a constant $C_1>0$ such that $$\label{3.10} |I(u_{j})|\leq C_1,\quad \|I'(u_{j})\|_{E^{*}}\leq C_1$$ for every $j\in \mathbb{N}$. We firstly prove that $\{u_{j}\}_{j\in \mathbb{N}}$ is bounded in $E$. By (\ref{3.1}), (\ref{3.3}), (H2) and the H\"{o}lder inequality, we have \label{3.11} \begin{aligned} \big(\frac{\mu}{2}-1\big)\|u_j\|^2 &= \mu I(u_{j})-I'(u_{j}) u_j\\ &\quad + \int_{\mathbb{R}}\big(\mu W(t,u_j(t)) -\big(W_{q}(t,u_j(t)),u_j(t)\Big)\big)dt\\ &\quad - \big(\frac{\mu}{2}-1\big)\int_{\mathbb{R}}\big(A u_j(t),\dot u_j(t)\big)dt\\ &\leq \mu I(u_{j})-I'(u_{j}) u_j+\big(\frac{\mu}{2}-1\big)\frac{\|A\|}{\sqrt{\beta}}\|u_j\|^2. \end{aligned} Combining this inequality with (\ref{3.10}), we obtain $$\label{3.12} \big(\frac{\mu}{2}-1\big)\big(1-\frac{\|A\|}{\sqrt{\beta}}\big)\|u_j\|^2 \leq \mu I(u_{j})-I'(u_{j}) u_j \leq \mu C_1+ C_1 \|u_{j}\|.$$ Since $\mu>2$ and $\|A\|<\sqrt{\beta}$, the inequality (\ref{3.12}) shows that $\{u_{j}\}_{j\in \mathbb{N}}$ is bounded in $E$. By Lemma \ref{lem 2.1}, the sequence $\{u_{j}\}_{j\in \mathbb{N}}$ has a subsequence, again denoted by $\{u_{j}\}_{j\in \mathbb{N}}$, and there exists $u\in E$ such that \begin{gather*} u_{j}\rightharpoonup u, \quad \text{weakly in } E,\\ u_{j}\to u,\quad \text{strongly in } L^2(\mathbb{R},\mathbb{R}^{n}). \end{gather*} Hence $$\big(I'(u_{j})-I'(u)\big) (u_{j}-u)\to 0,$$ and by Lemma \ref{lem 2.2} and the H\"{o}lder inequality, we have $$\int_{\mathbb{R}}\Big(W_{q}(t,u_{j}(t))-W_{q}(t,u(t)),u_{j}(t)-u(t)\Big)dt\to 0,$$ and $$\big|\int_{\mathbb{R}}\big(A \dot u_j(t)-A\dot u(t), u_j(t)-u(t)\big)dt\big|\leq \|A\|\|\dot u_j-\dot u\|\|u_j-u\|_2\to 0$$ as $j\to+\infty$. On the other hand, an easy computation shows that \begin{align*} &\big(I'(u_{j})-I'(u),u_{j}-u\big)\\ &= \|u_{j}-u\|^2 -\int_{\mathbb{R}}\big(A \dot u_j(t)-A\dot u(t), u_j(t)-u(t)\big)dt\\ &\quad -\int_{\mathbb{R}}\Big(W_{q}(t,u_{j}(t))-W_{q}(t,u(t)),u_{j}(t)-u(t)\Big)dt. \end{align*} Consequently, $\|u_{j}-u\| \to 0$ as $j\to +\infty$. \end{proof} Now we can give the proof of Theorem \ref{Thm 1.1}, we divide the proof into several steps. \subsection*{Proof of Theorem \ref{Thm 1.1}} \quad \noindent\textbf{Step 1} It is clear that $I(0)=0$ by Remark \ref{rem 1.1} and $I\in C^1(E,\mathbb{R})$ satisfies the (PS) condition by Lemmas \ref{lem 3.0} and \ref{lem 3.1}. \noindent \textbf{Step 2} We now show that there exist constants $\rho>0$ and $\alpha>0$ such that $I$ satisfies the condition (A1) of Lemma \ref{lem 2.4}. By (\ref{1.2}), for all $\varepsilon>0$, there exists $\delta>0$ such that $W(t,q)\leq \varepsilon |q|^2$ whenever $|q|\leq \delta$. Letting $\rho=\frac{\delta}{C}$ and $\|q\|=\rho$, we have $\|q\|_{\infty}\leq\delta$, where $C>0$ is defined in (\ref{2.1}). Hence $W(t,q(t))\leq \varepsilon |q(t)|^2$ for all $t\in \mathbb{R}$. Integrating on $\mathbb{R}$, we get $$\int_{\mathbb{R}}W(t,q(t))dt\leq \varepsilon \|q\|_2^2\leq \frac{\varepsilon}{\beta} \|q\|^2.$$ In consequence, combining this with (\ref{3.1}), we obtain that, for $\|q\|=\rho$, \label{3.13} \begin{aligned} I(q)&= \frac{1}{2}\|q\|^2+\frac{1}{2}\int_{\mathbb{R}}\big(A q(t), \dot q(t)\big)dt-\int_{\mathbb{R}}W(t,q(t))dt\\ &\geq \frac{1}{2}\|q\|^2-\frac{1}{2}\frac{\|A\|}{\sqrt{\beta}}\|q\|^2 -\frac{\varepsilon}{\beta} \|q\|^2\\ &= \big(\frac{1}{2}-\frac{1}{2}\frac{\|A\|}{\sqrt{\beta}} -\frac{\varepsilon}{\beta}\big)\|q\|^2. \end{aligned} Setting $\varepsilon=\frac{1}{4\beta}(1-\frac{\|A\|}{\sqrt{\beta}})$, the inequality (\ref{3.13}) implies $$I|_{\partial B_\rho}\geq\frac{1}{4} \big(1-\frac{\|A\|}{\sqrt{\beta}}\big)\frac{\delta^2}{C^2}=\alpha>0.$$ \noindent \textbf{Step 3} It remains to prove that there exists $e\in E$ such that $\|e\|>\rho$ and $I(e)\leq 0$, where $\rho$ is defined Step 2. By (\ref{3.1}), we have, for every $m\in \mathbb{R}\setminus \{0\}$ and $q\in E\setminus\{0\}$, \begin{align*} I(m\, q)&= \frac{m^2}{2}\|q\|^2 +\frac{m^2}{2} \int_{\mathbb{R}}\big(A q(t), \dot q(t)\big)dt-\int_{\mathbb{R}}W(t,m \,q(t))dt\\ &\leq \frac{m^2}{2}\big(1+\frac{\|A\|}{\sqrt{\beta}}\big)-\int_{\mathbb{R}}W(t,m \,q(t))dt. \end{align*} Take some $Q\in E$ such that $\|Q\|=1$. Then there exists a subset $\Omega$ of positive measure of $\mathbb{R}$ such that $Q(t)\neq 0$ for $t\in \Omega$. Take $m>0$ such that $m |Q(t)|\geq 1$ for $t\in \Omega$. Then, by (H5) and Remark \ref{rem 2.1}, we obtain that $$\label{3.14} I(m\, Q)\leq \frac{m^2}{2}\big(1+\frac{\|A\|}{\sqrt{\beta}}\big) -m^{\mu}\int_{\Omega}\alpha_0(t) |Q(t)|^{\mu}dt.$$ Since $\alpha_0(t)>0$ and $\mu>2$, (\ref{3.14}) implies that $I(m Q)<0$ for some $m>0$ such that $m |Q(t)|\geq 1$ for $t\in \Omega$ and $\|m Q\|> \rho$, where $\rho$ is defined in Step 2. By Lemma \ref{lem 2.4}, $I$ possesses a critical value $c\geq\alpha>0$ given by $$c=\inf_{g\in\Gamma}\max_{s\in[0,1]}I(g(s)),$$ where $$\Gamma=\{g\in C([0,1], E): g(0)=0,\,g(1)=e\}.$$ Hence there is $q\in E$ such that $I(q)=c$, $I'(q)=0$. \smallskip \noindent \textbf{Step 4} Now suppose that $W(t,q)$ is even in $q$; i.e., (H6) holds, which implies that $I$ is even. Furthermore, we already know that $I(0)=0$ and $I\in C^1(E,\mathbb{R})$ satisfies the (PS) condition in Step 1. To apply Lemma \ref{lem 2.6}, it suffices to prove that $I$ satisfies the conditions (A3) and (A4) of Lemma \ref{lem 2.6}. Here we take $V=\{0\}$ and $X=E$. (A3) is identically the same as in Step 2, so it is already proved. Now we prove that (A4) holds. Let $\tilde{E}\subset E$ be a finite dimensional subspace. 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