\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 11, pp. 1--16.\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-2013/11\hfil Infinitely many homoclinic orbits] {Existence of infinitely many homoclinic orbits for second-order systems involving Hamiltonian-type equations} \author[A. Daouas, A. Moulahi \hfil EJDE-2013/11\hfilneg] {Adel Daouas, Ammar Moulahi } % in alphabetical order \address{Adel Daouas \newline Mathematics department, College of sciences, Taibah University, Saudi Arabia} \email{adaouas@taibahu.edu.sa} \address{Ammar Moulahi \newline College of Business and Economics, Qassim University, Saudi Arabia} \email{ammar.moulahi@fsm.rnu.tn} \thanks{Submitted March 25, 2012. Published January 14, 2013.} \subjclass[2000]{34C37, 37J45, 70H05} \keywords{Homoclinic solutions; differential system; critical point} \begin{abstract} We study the second-order differential system $$\ddot u + A\dot{u}- L(t)u+ \nabla V(t,u)=0,$$ where $A$ is an antisymmetric constant matrix and $L \in C(\mathbb{R}, \mathbb{R}^{N^2})$. We establish the existence of infinitely many homoclinic solutions if $W$ is of subquadratic growth as $|x| \to +\infty$ and $L$ does not satisfy the global positive definiteness assumption. In the particular case where $A=0$, earlier results in the literature are generalized. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \newcommand\cb{\mathbb C} \newcommand\zb{\mathbb Z} \def\a{\alpha} \def\C{\cal C} \def\b{\beta} \def\e{\eta} \def\g{\gamma} \def\s{\sigma} \def\d{\delta} \def\L{\Lambda} \def\l{\lambda} \def\nb{\mathbb N} \def \ds{\displaystyle} \let\w=\wedge \let\t=\theta \let\L=\to \def\v{\varphi} \def \C{\cal C} \let\l=\rightarrow \let\ep=\varepsilon \let\s=\subset \let\O=\Omega \section{Introduction} Let us consider the second-order differential system $$\ddot u + A\dot{u}- L(t)u + \nabla V(t,u)=0,\label{DS}$$ where $A$ is an antisymmetric constant matrix with small size in $\mathbb{R}^{2N}$ (see the estimation \eqref{e2.2}), $L \in C(\mathbb{R}, \mathbb{R}^{N^2})$ is a symmetric matrix valued function and $V \in C(\mathbb{R}\times\mathbb{R}^N, \mathbb{R})$ is of class $C^1$ in the second variable. We will say that a solution $u$ of \eqref{DS} is \textit{homoclinic} (to 0) if $u \in C^2(\mathbb{R}, \mathbb{R}^N)$, $u(t)\to 0$ and $\dot{u}(t)\to 0$ as $t\to \pm\infty$. For the particular case $A=0$, \eqref{DS} is just the Hamiltonian system $$\ddot u - L(t)u + \nabla V(t,u)=0.\label{HS}$$ In recent years, existence and multiplicity of homoclinic solutions for the second order Hamiltonian system \eqref{HS} have been investigated by many authors via the critical point theory, see \cite{a1}--\cite{r2}, \cite{s1}--\cite{z4} and references therein. Most of them treat the superquadratic case under the so-called global Ambrosetti-Rabinowitz condition; that is, there exists $\mu >2$ such that $$0 < \mu V(t,x) \le (\nabla V(t,x),x),\quad \text{for all } (t, x)\in \mathbb{R} \times \mathbb{R}^N\backslash \{0\}.$$ Exceptionally, in \cite{d3}, the author considered, in part of the paper, the case where the potential is of subquadratic growth as $|x| \to +\infty$. Moreover, contrary to the previous works, he removed the global positive definiteness of the matrix $L(t)$ by assuming \begin{itemize} \item[(L1)] for the smallest eigenvalue of $L(t)$, i.e., $l(t)= \inf_{|x|=1} (L(t)x , x)$, there exists a constant $\a <1$ such that $$l(t)|t |^{\a -2}\to \infty\quad \text{as}\ |t|\to \infty,$$ \item[(L2)] for some positive constants $a,r$, one of the following is true: \begin{itemize} \item[(i)] $L \in C^1(\mathbb{R},\mathbb{R}^{N^2})$ and $|L'(t)x| \le a|L(t)x|$ for all $|t | > r$ and all $x \in \mathbb{R}^{N}$ with $|x| = 1$, or \item[(ii)] $L \in C^2(\mathbb{R},\mathbb{R}^{N^2})$ and $(aL(t)x- L''(t)x, x) \ge 0$ for all $|t | > r$ and all $x \in \mathbb{R}^{N}$ with $|x| = 1$, \end{itemize} where $L'(t) = (d/dt)L(t), L''(t) = (d^2/dt^2)L(t)$. \end{itemize} Under other suitable conditions he established the existence and multiplicity of homoclinic solutions for \eqref{HS}. Later, his results were partially improved in \cite{y1,z1}. Recently, the authors in \cite{z2,z3}, treated the special case where $V(t,x)= a(t)|x|^\mu$ with $1< \mu<2$ and $L(t)$ is a positive definite matrix for all $t\in\mathbb{R}$. They proved the existence of a nontrivial homoclinic solution for \eqref{HS} and \eqref{DS} respectively; where the system \eqref{DS} was considered for the first time. Later, multiplicity of homoclinics for \eqref{HS} was studied in \cite{s1} for the same class of Hamiltonians. However, in mathematical physics, it is of frequent occurrence in \eqref{HS} that the global definiteness of $L(t)$ is not satisfied (see \cite{d3} for an example). As far as the authors know, there is no research concerning the existence and multiplicity of homoclinic solutions for \eqref{DS} apart from \cite{z3}. In this paper, motivated by \cite{d3,z3} mainly, we study the existence of infinitely many homoclinic solutions for \eqref{DS} in the case where $L$ does not satisfy the global positive definiteness assumption. Also, the potential $V$ will be of subquadratic growth as $|x|\to +\infty$ and is not necessarily of the form $V(t,x)= a(t)|x|^\mu$. In the first result we assume that $V(t,x)=a(t)W(x)$ with $W \in C^1(\mathbb{R}^N, \mathbb{R})$, $a \in C(\mathbb{R}, \mathbb{R})\cap L^2(\mathbb{R}, \mathbb{R})$ are nonnegative functions and $a \not\equiv 0$. The difficulty in studying this class of nonlinearities comes essentially from the fact that $\inf_{t\in \mathbb{R} } a(t)=0$ and then there is no constant $b> 0$ such that $V(t,x) \ge b |x|^\gamma$ for all $t\in \mathbb{R}$, which essential in previous works. Moreover, in the case where $A\not=0$, we are unable to verify the Palais-Smale condition. To overcome this obstacle, we use a variant fountain theorem established in \cite{z5}. For our first theorem use the following assumptions: \begin{itemize} \item[(L3)] $0\notin \sigma\big(-(d^2/dt^2) + L(t)- A (d/dt)\big)$, \item[(V1)] $W(0)=0$ and there exist positive constants $a_1, a_2, r$ and $1\le \gamma\le\mu< 2$ such that $$a_1 |x|^\gamma \le W(x)\le a_2 |x|^\mu,\quad \text{for all } |x| \ge r,$$ \item[(V2)] there exist positive constants $a_3, \omega$ and $\nu\in [1,2)$ such that $$W(x)\ge a_3 |x|^\nu,\quad \text{for all} \ |x| \le \omega,$$ \item[(V3)] there exist constants $a_4> 0$ and $\beta \in[1,2)$ such that $$|\nabla W(x)|\le a_4( |x|^{\beta-1}+ 1) \quad \text{for all } x \in \mathbb{R}^N,$$ \item[(V4)] $W$ is even, \item[(V5)] $a\in L^{2}(\mathbb{R},\mathbb{R})$ and $\operatorname{meas}\{t\in \mathbb{R}: a(t)=0\}=0$. \end{itemize} \begin{theorem} \label{thm1.1} Assume that $L$ satisfies {\rm (L1)--(L3)} and $V$ satisfies {\rm (V1)--(V5)}. Then system \eqref{DS} has infinitely many homoclinic solutions. \end{theorem} In the particular for the case $A=0$, we have the following result. \begin{corollary} \label{coro1.2} Under the assumptions of Theorem \ref{thm1.1}, system \eqref{HS} has infinitely many homoclinic solutions. \end{corollary} \begin{remark} \label{rmk.13} \rm Consider $$L(t)=(t^2-1)I_N\quad\text{and}\quad V(t, x) = \frac{|\sin t|}{ |t|+1}|x|^{5/4}\log( 1+|x|^{1/2}).$$ A straightforward computation shows that $L$ and $V$ satisfy the conditions of Theorem \ref{thm1.1} but since $\inf_{t\in\mathbb{R}} V(t, x)= 0$, the assumptions of \cite[Theorem 1.2]{d3} and \cite[Theorem 1.1]{z1} do not hold. So, in some sense, Corollary \ref{coro1.2} completes the corresponding results in \cite{d3,z1} and the one in \cite{z4} for the case $\beta=0$. Moreover, Theorem \ref{thm1.1} generalizes the result of \cite{z3}. \end{remark} Our second main result concerns a class of nonlinearities with bounded gradient which cover the functions of the type $V(t,x)= Ln( 1+ |x|^{3/2})$ and which not necessarily of the form $V(t,x)= a(t) W(x)$. Homoclinic solutions to \eqref{HS} for this class of Hamiltonians was investigated in \cite[Theorem 1.3]{d3} under the assumption of positive definiteness of $L(t)$. Here, we omit this condition mainly. Precisely we have the following assumptions: \begin{itemize} \item[(V1')] $V(t,0)\equiv0$ and $V(t,x)\to\infty$ as $|x|\to\infty$ uniformly in $t\in\mathbb{R}$, \item[(V2')] there exist constants $a_1, \omega > 0$ and $\nu\in [1,2)$ such that $$V(t, x)\ge a_1 |x|^\nu,\quad \text{for all } |x| \le \omega, t\in \mathbb{R},$$ \item[(V3')] there exists a constant $\ M > 0$ such that $$|\nabla V(t,x)|\le M,\quad \text{for all } (t, x) \in \mathbb{R}\times\mathbb{R}^N,$$ \item[(V4')] there exist constants $a_2, r > 0$ and $\beta\in [1/2, 1)$ such that $$|\nabla V(t, x)|\le a_2 |x|^\beta,\quad \text{for all } |x| \le r, \; t\in \mathbb{R},$$ \item[(V5')] $V(t,-x)=V(t,x)\ge 0,\quad\text{for all } (t, x) \in \mathbb{R}\times\mathbb{R}^N$. \end{itemize} \begin{theorem} \label{thm1.4} Assume that $L$ satisfies {\rm (L1)--(L3)} and $V$ satisfies {\rm (V1')--(V5')}. Then system \eqref{DS} has infinitely many homoclinic solutions. \end{theorem} \begin{corollary} \label{coro1.5} Under the assumptions of Theorem \ref{thm1.4}, system \eqref{HS} has infinitely many homoclinic solutions. \end{corollary} \begin{remark} \label{rmk1.6} \rm Consider the function $$V(t, x)= \log( 1+|x|^{3/2}).$$ A straightforward computation shows that $V$ satisfies the conditions of Theorem \ref{thm1.4} but does not satisfy condition (W4) in \cite[Theorem 1.1]{z1}. Moreover, since $L(t)$ is unnecessarily positive definite, Corollary \ref{coro1.5} improves the corresponding results in \cite{d3,z1}. \end{remark} \section{Preliminary results} We establish our results by using critical point theory, but we first give some preliminaries (for details see \cite{d3}). We denote by $B$ the selfadjoint extension of the operator $-(d^2/dt^2) + L(t)$ with the domain $\mathcal{D}(B)\subset L^2\equiv L^2(\mathbb{R}, \mathbb{R}^N)$. Let $|B|$ be the absolute value of $B$ and $|B|^{1/2}$ be the square of $|B|$. Let $E=\mathcal{D}(|B|^{1/2})$, the domain of $|B|^{1/2}$, and define on $E$ the inner product $$(u,v)_0= (|B|^{1/2}u, |B|^{1/2}v)_{L^2} + (u,v)_{L^2}$$ and norm $$\|u\|_0= (u,u)_0^{1/2},$$ where $(.,.)_{L^2}$ denotes the inner product of $L^2$. Then $E$ is a Hilbert space. It is easy to prove that the spectrum $\sigma(B)$ consists of eigenvalues numbered in $\lambda_{1} \le \lambda_{2} \le \dots\nearrow\infty$ (counted with their multiplicities), and a corresponding system of eigenfunctions $\{e_i\}_{i\in\mathbb{N}}$ of $B$ forms an orthonormal basis in $L^2$. Define $$n^- = \#\{i : \lambda_{i} < 0\}, \quad n^0 = \#\{i : \lambda_{i} = 0\}, \quad \bar{n} = n^- + n^0\label{e2.1}$$ and \begin{gather*} E^- = \operatorname{span}\{e_1, \dots ,e_{n^-}\}, \quad E^0 = \operatorname{span}\{e_{n^- +1}, \dots ,e_{\bar{n}}\} = \ker B,\\ E^+ =\overline{ \operatorname{span}\{e_{\bar{n} +1}, \dots \}}. \end{gather*} Then one has the orthogonal decomposition $E = E^- \oplus E^0 \oplus E^+$ with respect to the inner product $(\cdot,\cdot)_0$. Now we introduce on $E$ the following inner product and norm: $$(u, v) =(|B|^{1/2}u, |B|^{1/2}v)_{L^2} + (u^0,v^0)_{L^2}$$ and $$\|u\|= (u,u)^{1/2},$$ where $u = u^- +u^0 +u^+$ and $v = v^- +v^0 +v^+ \in E = E^- \oplus E^0 \oplus E^+$. Clearly the norms $\|\cdot\|_0$ and $\|\cdot\|$ are equivalent (see \cite{d3}). Furthermore, the decomposition $E = E^- \oplus E^0 \oplus E^+$ is orthogonal with respect to the inner products $(\cdot,\cdot)$ and $(\cdot,\cdot)_{L^2}$. For the rest of this article, $\|\cdot\|$ will be the norm used on $E$. The following fact on $E$ will be needed. \begin{lemma}[\cite{d3}] \label{lem2.1} Suppose that $L(t)$ satisfies {\rm (L1)}. Then $E$ is continuously embedded in $W^{1,2}(\mathbb{R}, \mathbb{R}^N)$, and consequently there exists $\delta > 0$ such that $$\|u\| _{W^{1,2}(\mathbb{R}, \mathbb{R}^N)} \le \delta\|u\|,\quad \text{for all}\ u \in E,$$ where $\|u\| _{W^{1,2}(\mathbb{R}, \mathbb{R}^N)} =(\|u\|^2 _{L^2} + \|\dot{u}\|^2 _{L^2})^{1/2}$. \end{lemma} Now, we make the following estimation on the norm of the matrix $A$, $$|A|< \frac{1}{\delta^2},\label{e2.2}$$ where $|\cdot|$ is the standard norm of $\mathbb{R}^{N^2}$. Moreover, using (V5), we note that $a$ is bounded and can be seen as a weight function. So, for $p\ge 1$, the weighted norm $\|\cdot\|_{L^p(a)}$ will be defined on $E$ by $$\|u\|_{L^p(a)}=\Big[\int_\mathbb{R} a(t)|u(t)|^p dt\Big]^{1/p}.$$ From \cite[Lemmas 2.2 and 2.3]{d3}, we have the following two lemmas. \begin{lemma}[\cite{d3}] \label{lem2.2} Suppose that $L(t)$ satisfies {\rm (L1)}. Then $E$ is compactly embedded in $L^p$ for any $1\le p \le \infty$, which implies that there exists a constant $C_p >0$ such that $$\|u\| _{L^p}\le C_p\|u\|, \quad \text{for all } u \in E.\label{e2.3}$$ \end{lemma} \begin{lemma}[\cite{d3}] \label{lem2.3} Suppose that $L(t)$ satisfies {\rm (L1), (L2)}. Then $\mathcal{D}(B)$ is continuously embedded in $W^{2,2}(\mathbb{R}, \mathbb{R}^N)$, and consequently, we have $$|u(t)|\to 0\quad \text{and}\quad |\dot{u}(t)|\to 0\quad \text{as } |t|\to \infty,$$ for all $u\in \mathcal{D}(B)$. \end{lemma} \begin{lemma} \label{lem2.4} Suppose assumption {\rm (V5)} holds. If $q_k \rightharpoonup q$ (weakly) in $E$, then $\nabla V(t,q_k)\to \nabla V(t,q)$ in $L^2(\mathbb{R}, \mathbb{R}^N)$. \end{lemma} \begin{proof} Assume that $q_k \rightharpoonup q$ in $E$. By the Banach-Steinhaus Theorem the sequence $(q_k)_{k\in \mathbb{N}}$ is bounded in E and by \eqref{e2.3}, there exists a constant $d_1> 0$ such that $$\sup_{k\in \mathbb{N}} \|q_k\|_{L^\infty} \le d_1, \quad \|q\|_{L^\infty}\le d_1. \label{e2.4}$$ Since $\nabla W$ is continuous, by \eqref{e2.4} there exists a constant $d_2 >0$ such that $$|\nabla W (q_k (t))|\le d_2, \quad |\nabla W (q(t))|\le d_2,$$ for all $k \in \mathbb{N}$ and $t\in \mathbb{R}$. Hence, $$|\nabla V (t,q_k(t))-\nabla V (t,q(t))|\le 2d_2 a(t).$$ On the other hand, by Lemma \ref{lem2.2}, $q_k\to q$ in $L^2$, passing to a subsequence if necessary, we obtain $q_k\to q$ for almost every $t\in \mathbb{R}$. Then, using (V5), the Lebesgue's Convergence Theorem gives the conclusion. \end{proof} 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 $C^1$-functional $\Phi_\lambda : E\to \mathbb{R}$ defined by $$\Phi_\lambda (u):= \mathcal{A} (u) - \lambda \mathcal{B} (u), \quad \lambda \in [1,2].$$ \begin{theorem}[{\cite[Theorem 2.2]{z5}}] \label{thm2.5} Assume that the functional $\Phi_\lambda$ defined above satisfies \begin{itemize} \item[(T1)] $\Phi_\lambda$ maps bounded sets to bounded sets uniformly for $\lambda \in [1,2]$. Moreover, $\Phi_\lambda (-u)=\Phi_\lambda (u)$ for all $(\lambda, u) \in [1,2]\times E$, \item[(T2)] $\mathcal{B}(u)\ge 0$ for all $z\in E; \mathcal{B}(u)\to\infty$ as $|z|\to \infty$ on any finite dimensional subspace of $E$, \item[(T3)] there exist $\rho_k> r_k>0$ such that $$a_k(\lambda):=\inf_{u\in Z_k, \|u\|=\rho_k }\Phi_\lambda (u) \ge 0> b_k(\lambda):=\max_{u\in Y_k, \|u\|=r_k} \Phi_\lambda (u),$$ for all $\lambda \in [1,2]$, and $$d_k(\lambda):=\inf_{u\in Z_k, \|u\|\le\rho_k }\Phi_\lambda (u)\to 0\quad \text{as k\to\infty, uniformly for } \lambda \in [1,2].$$ \end{itemize} Then there exist $\lambda_n \to 1, u_{\lambda_n} \in Y_n$ such that $$\Phi'_{\lambda_n} |_{Y_n}(u_{\lambda_n})=0, \quad \Phi_{\lambda_n} (u_{\lambda_n})\to f_k \in [d_k(2), b_k(1)]\quad \text{as } n\to \infty.$$ Particularly, if $\{u_{\lambda_n}\}$ has a convergent subsequence for every $k\in \mathbb{N}$, then $\Phi_1$ has infinitely many nontrivial critical points $\{u_k\} \in E\backslash \{0\}$ satisfying $\Phi_1(u_k)\to 0^-$ as $k\to\infty$. \end{theorem} \section{Proof of Theorem \ref{thm1.1}} Let $\Phi$ be the functional defined on $E$ by \begin{aligned} \Phi(u)&=\frac{1}{2}\int_\mathbb{R} \Big[|\dot u(t)|^2 +(L(t)u(t),u(t))\Big]dt +\frac{1}{2}\int_\mathbb{R}(Au(t),\dot{u}(t))dt -\int_\mathbb{R} V(t,u(t))dt\\ &=\frac{1}{2} \Big(\|u^+\|^2-\|u^-\|^2\Big) +\frac{1}{2}\int_\mathbb{R} (Au(t),\dot{u}(t))dt -\int_\mathbb{R} V(t,u(t))dt, \label{e3.1} \end{aligned} for all $u = u^- +u^0 +u^+ \in E = E^- \oplus E^0 \oplus E^+$. \begin{lemma} \label{lem3.1} Under the conditions of Theorem \ref{thm1.1}, $\Phi\in C^1(E,\mathbb{R})$ and \begin{align*} \Phi'(u)v &=\int_\mathbb{R} \Big[(\dot u(t),\dot v(t))+(L(t)u(t),v(t))\Big]dt +\int_\mathbb{R} (Au(t),\dot{v}(t))dt \\ &\quad -\int_\mathbb{R}(\nabla V(t,u(t)),v(t)) dt. \end{align*} for all $u = u^- + u^0 +u^+$, $v = v^- + v^0 +v^+$ in $E = E^- \oplus E^0 \oplus E^+$. Moreover, any critical point of $\Phi$ on $E$ is a homoclinic solution of \eqref{DS}. \end{lemma} \begin{proof} Rewrite $\Phi= \Psi_1+\Psi_2-\Psi_3$ where \begin{gather*} \Psi_1(u):=\frac{1}{2}\int_\mathbb{R} \Big[|\dot u(t)|^2 +(L(t)u(t),u(t))\Big]dt,\quad \Psi_2(u):=\frac{1}{2}\int_\mathbb{R}(Au(t),\dot{u}(t))dt,\\ \Psi_3(u):= \int_\mathbb{R} V(t,u(t))dt. \end{gather*} It is known \cite{d3} that $\Psi_1\in C^1(E,\mathbb{R})$ and for all $u,v \in E$, $$\Psi'_1(u)v= \int_\mathbb{R} \Big[(\dot u(t),\dot v(t))+(L(t)u(t),v(t))\Big]dt.$$ Also, we have $\Psi_2\in C^1(E,\mathbb{R})$, and for all $u,v \in E$, $$\Psi_2'(u)v=\int_\mathbb{R} (Au(t),\dot{v}(t))dt.$$ Indeed, using Lemma \ref{lem2.1}, the quadratic form $\Psi_2$ is continuous and therefore it is of class $C^1$. Furthermore, by the use of the antisymmetric property of $A$, we obtain the result. It remains to show that $\Psi_3\in C^1(E,\mathbb{R})$ and for all $q,v \in E$, $$\Psi_3'(q)v=\int_\mathbb{R} (\nabla V(t,q(t)),v(t))dt.$$ Fix $q \in E$, let $c_1= \sup_{|x|\le \|q\|_{L^\infty}} |\nabla W(x)|$ and define $J(q) : E\to \mathbb{R}$ as follows $$J(q)v =\int_\mathbb{R} (\nabla V(t,q(t)),v(t))dt,\quad \forall v \in E.$$ Then $J(q)$ is linear and bounded. Indeed, $$|\nabla V(t,q(t))|= a(t)|\nabla W(q(t))|\le c_1 a(t), \quad\forall t \in \mathbb{R}$$ and by \eqref{e2.3}, we obtain \begin{aligned} |J(q)v|&=|\int_\mathbb{R} (\nabla V(t,q(t)),v(t))dt|\\ &\le c_1\int_\mathbb{R} a(t) |v(t)|dt\\ & \le c_1 \|a\|_2 \|v\|_2\\ &\le c_1 C_2\|a\|_2 \|v\|. \end{aligned} \label{e3.2} Moreover, for $q, v \in E$, by the Mean Value Theorem, we have $$\int_\mathbb{R} V(t,q(t)+ v(t))dt- \int_\mathbb{R} V(t,q(t))dt =\int_\mathbb{R} (\nabla V(t,q(t)+ h(t)v(t)), v(t))dt,$$ where $h(t) \in (0,1)$. Also, by Lemma \ref{lem2.4} and the H\"{o}lder inequality, we have \begin{aligned} &\int_\mathbb{R} (\nabla V(t,q(t)+ h(t)v(t)), v(t))dt - \int_\mathbb{R} (\nabla V(t,q(t)), v(t))dt\\ &=\int_\mathbb{R} (\nabla V(t,q(t)+ h(t)v(t))-\nabla V(t,q(t)),v(t)) dt\to 0, \end{aligned} \label{e3.3} as $v\to 0$ in $E$. Combining \eqref{e3.2} and \eqref{e3.3} we obtain the result. Now, we prove that $\Psi'_3$ is continuous. Suppose that $q\to q_0$ in $E$ and note that $$\Psi'_3(q)v-\Psi'_3(q_0)v= \int_\mathbb{R} (\nabla V(t,q(t)) -\nabla V(t,q_0(t)), v(t))dt.$$ By Lemma \ref{lem2.4} and the H\"{o}lder inequality, we obtain $$\Psi'_3(q)v-\Psi'_3(q_0)v \to 0, \quad \text{as } q\to q_0.$$ Now, we check that critical points of $\Phi$ are homoclinic solutions for \eqref{DS}. In fact, if $u$ is a critical point of $\Phi$, by Lemma \ref{lem3.1}, we have $L(t)u(t) -\nabla V(t,u(t))$ is the weak derivative of $\dot{u} + Au$. Since $L \in C(\mathbb{R}, \mathbb{R}^{N^2})$ and $V \in C^1(\mathbb{R}\times\mathbb{R}^N, \mathbb{R})$, we see that $\dot{u} + Au$ is continuous and consequently $\dot{u}$ is continuous which yields $u \in C^2(\mathbb{R}, \mathbb{R}^{N})$; i.e., $u$ is a classical solution of \eqref{DS}. Finally, to prove that $\dot{u}(t)\to 0$ as $|t|\to \infty$, note that by Lemma \ref{lem2.3} it suffices to show that any critical point of $\Phi$ on $E$ is an element of $\mathcal{D}(B)$. Indeed, by Lemma \ref{lem2.1}, we know that $u\in W^{1,2}(\mathbb{R}, \mathbb{R}^N)$ and hence $u(t)\to 0$ as $|t|\to \infty$. Moreover, since $W \in C^1(\mathbb{R}^N, \mathbb{R})$, there exists $d>0$ such that $$|\nabla W(u(t))|\le d,\quad \forall t\in \mathbb{R}.\label{e3.4}$$ From \eqref{DS} and this inequality, we receive \begin{aligned} \|Bu\|_{L^2}^2 &=\|A\dot{u}+ \nabla V(t,u)\|_{L^2}^2\\ &\le 2\int_\mathbb{R} | A\dot{u}(t)|^2 dt +2 d^2\int_\mathbb{R} |a(t)|^2 dt. \end{aligned}\label{e3.5} By \eqref{e3.5} and the fact $|\dot{u}|, a \in L^2(\mathbb{R},\mathbb{R})$ one sees that $\|Bu\|_{L^2} < \infty$; i.e., $u \in \mathcal{D}(B)$. \end{proof} To apply Theorem \ref{thm2.5} for proving Theorem \ref{thm1.1}, we define the functionals $\mathcal{A}, \mathcal{B}$ and $\Phi_\lambda$ on the space $E$ by \begin{gather*} \mathcal{A} (u)=\frac{1}{2} \|u^+\|^2 +\frac{1}{2}\int_\mathbb{R} (Au(t), \dot{u}(t))dt,\quad \mathcal{B} (u)= \frac{1}{2} \|u^-\|^2 +\int_\mathbb{R} V(t,u(t))dt, \\ \begin{aligned} \Phi_\lambda (u)&:= \mathcal{A} (u) - \lambda \mathcal{B} (u)\\ &=\frac{1}{2} \|u^+\|^2 +\frac{1}{2}\int_\mathbb{R} (Au(t),\dot{u}(t))dt -\lambda \Big(\frac{1}{2} \|u^-\|^2 +\int_\mathbb{R} V(t,u(t))dt\Big) \end{aligned} \end{gather*} for all $u = u^- +u^0 +u^+$ in $E = E^- \oplus E^0 \oplus E^+$ and $\lambda \in[1,2]$. From Lemma \ref{lem3.1}, we know that $\Phi_\lambda \in C^1(E,\mathbb{R})$ for all $\lambda \in[1,2]$. Let $X_j= span \{e_j\}$ for all $j\in \mathbb{N}$, where $\{e_n; n\in \mathbb{N}\}$ is the system of eigenfunctions given below. Note that $\Phi_1=\Phi$, where $\Phi$ is the functional defined in \eqref{e3.1}. \begin{lemma} \label{lem3.2} Under the assumption {\rm (V1)}, we have $\mathcal{B}(u)\ge 0$ and $\mathcal{B}(u)\to\infty$ as $\|u\| \to \infty$ on any finite dimensional subspace of $E$. \end{lemma} \begin{proof} Since $a$ and $W$ are nonnegative it is obvious, by the definition of $\mathcal{B}$, that $\mathcal{B}(u)\ge 0$. We claim that for any finite dimensional subspace $F\subset E$, there exists $\epsilon >0$ such that $$\operatorname{meas}\big(\{t\in \mathbb{R} : a(t)|u(t)|^\gamma\ge \epsilon \|u\|^\gamma\}\big)\ge\epsilon, \quad \forall u \in F\backslash\{0\}. \label{e3.6}$$ If not, for any $n\in \mathbb{N}$, there exists $u_n\in F\backslash\{0\}$ such that $$\operatorname{meas}(\{t\in \mathbb{R} :a(t) |u_n(t)|^\gamma\ge \frac{1}{n} \|u_n\|^\gamma\})< \frac{1}{n}.$$ Let $v_n:= \frac{u_n}{ \|u_n\|}$. Then $v_n \in F$, $\|v_n\|=1$ for all $n\in \mathbb{N}$ and $$\operatorname{meas}(\{t\in \mathbb{R} :a(t)|v_n(t)|^\gamma\ge \frac{1}{n}\})< \frac{1}{n},\quad \forall n\in \mathbb{N}.\label{e3.7}$$ Passing to a subsequence if necessary, we may assume $v_n\to v_0$ in $E$ for some $v_0 \in F$ since $F$ is of finite dimension. Evidently, $\|v_0\|=1$. By the equivalence of norms on $F$, we have $v_n\to v_0$ in $L^\gamma(a)$; i.e., $$\int_\mathbb{R} a(t)|v_n-v_0|^\gamma dt \to 0, \quad \text{as } n \to \infty.\label{e3.8}$$ Moreover, since $\|v_0\|_{L^\infty} >0$, by (V5) and the definition of $\|\cdot\|_{L^\infty}$, it is easy to see that there exists a constant $\delta_0 >0$ such that $$\operatorname{meas}(\{t\in \mathbb{R} ; a(t)|v_0(t)|^\gamma\ge \delta_0 \}) \ge \delta_0.\label{e3.9}$$ For any $n\in \mathbb{N}$, let $$\Lambda_n=\{t\in \mathbb{R} :a(t)|v_n(t)|^\gamma < \frac{1}{n}\}, \quad \Lambda_n^c=\mathbb{R} \backslash\Lambda_n=\{t\in \mathbb{R} : a(t)|v_n(t)|^\gamma\ge \frac{1}{n}\}.$$ \ Set $\Lambda_0=\{t\in \mathbb{R} :a(t)|v_0(t)|^\gamma\ge \delta_0 \}$. Then, for $n$ large enough, by \eqref{e3.7} and \eqref{e3.9}, we have $$\operatorname{meas}(\Lambda_n \cap \Lambda_0) \ge \operatorname{meas}(\Lambda_0)- \operatorname{meas}(\Lambda_n^c) \ge \delta_0-1/n\ge \delta_0/2.$$ Consequently, for $n$ large enough, there holds \begin{align*} \int_\mathbb{R} a(t)|v_n-v_0|^\gamma dt &\ge \int_ {\Lambda_n \cap \Lambda_0}a(t)|v_n-v_0|^\gamma dt\\ & \ge \frac{1}{2^{\gamma -1}} \Big(\int_ {\Lambda_n \cap \Lambda_0}a(t)|v_0|^\gamma dt -\int_ {\Lambda_n \cap \Lambda_0}a(t)|v_n|^\gamma dt\Big)\\ &\ge \frac{1}{2^{\gamma -1}}(\delta_0-{1/n}) \operatorname{meas}(\Lambda_n \cap \Lambda_0)\\ &\ge \frac{\delta_0^2}{2^{\gamma +1}}>0. \end{align*} This contradicts \eqref{e3.8} and therefore \eqref{e3.6} holds. For the $\epsilon$ given in \eqref{e3.6}. Let $$\Lambda_u=\{t\in \mathbb{R} :a(t)|u(t)|^\gamma\ge \epsilon \|u\|^\gamma\},\quad \forall u \in F\backslash\{0\}.$$ Then $$\operatorname{meas}(\Lambda_u)\ge \epsilon,\quad \forall u \in F\backslash\{0\}.\label{e3.10}$$ Observing that for $u\in F$ with $\|u\|\ge r (\|a\|_{L^\infty}/\epsilon)^{1/\gamma}$, there holds $$|u(t)|\ge r,\quad \forall t\in\Lambda_u. \label{e3.11}$$ Combining \eqref{e3.10}, \eqref{e3.11} and (V1), for any $u\in F$ with $\|u\|\ge r (\|a\|_{L^\infty}/\epsilon)^{1/\gamma}$, we obtain \begin{align*} \mathcal{B}(u)&=\frac{1}{2} \|u^-\|^2 +\int_\mathbb{R} V(t,u(t))dt\\& \ge\int_{\Lambda_u} V(t,u(t))dt\\& \ge a_1\int_{\Lambda_u} a(t)|u(t)|^\gamma dt\\ &\ge a_1\epsilon \|u\|^\gamma \operatorname{meas}(\Lambda_u) \ge a_1 \epsilon^2 \|u\|^\gamma , \end{align*} which implies that $\mathcal{B}(u)\to\infty$ as $\|u\| \to \infty$ on $F$. \end{proof} \begin{lemma} \label{lem3.3} Under the assumptions of Theorem \ref{thm1.1}, there exist a positive integer $k_0$ and a sequence $\rho_k \to 0^+$ as $k\to\infty$ such that $$a_k(\lambda):=\inf_{u\in Z_k, \|u\|=\rho_k }\Phi_\lambda (u) >0, \quad \forall k\ge k_0,$$ and $$d_k(\lambda):=\inf_{u\in Z_k, \|u\|\le\rho_k }\Phi_\lambda (u)\to 0\quad \text{as k\to\infty, uniformly for } \lambda \in [1,2],$$ where $Z_k=\overline{\oplus_{j=k}^\infty X_j}$. \end{lemma} \begin{proof} Note that $Z_k\subset E^+$ for all $k\ge \bar{n}+1$ where $\bar{n}$ is the integer defined in \eqref{e2.1}. So, for any $k\ge \bar{n}+1$ and $(\lambda, u)\in [1,2]\times Z_k$, we have \begin{aligned} \Phi_\lambda (u) &\ge \frac{1}{2} \|u\|^2 -\frac{1}{2} |A|\|u\|_{L^2}\|\dot{u}\|_{L^2} -2 \int_\mathbb{R} V(t,u(t))dt\\ &\ge \frac{1}{2} \Big(1 -\delta^2|A|\Big)\|u\|^2 -2 \int_\mathbb{R} V(t,u(t))dt, \label{e3.12} \end{aligned} with $1 -\delta^2|A| > 0$ by \eqref{e2.2}. On the other hand, by the mean value theorem and (V3), we have \begin{aligned} \int_\mathbb{R} V(t,u(t))dt &= \int_\mathbb{R} (\nabla V(t,\theta (t)u(t)), u(t))dt\\ &\le a_4\int_\mathbb{R} a(t)|u(t)|^{\beta }dt + a_4\int_\mathbb{R} a(t)|u(t)|dt \end{aligned} \label{e3.13} where $\theta (t)\in (0,1)$. Since the function $a$ is bounded, by \eqref{e3.13} there exists $c_1> 0$ such that $$\int_\mathbb{R} V(t,u(t))dt\le c_1 \Big(\|u\|_{L^\beta}^{\beta } + \|u\|_{L^1}\Big). \label{e3.14}$$ Combining \eqref{e3.12} and \eqref{e3.14}, we obtain $$\Phi_\lambda (u)\ge\frac{1}{2} \Big( 1 -\delta^2|A|\Big)\|u\|^2 -2c_1\Big(\|u\|_{L^\beta}^{\beta }+ \|u\|_{L^1}\Big). \label{e3.15}$$ For $k\in \mathbb{N}$, define $$l_1(k):= \sup_{u\in Z_k, \|u\|=1} \|u\|_{L^1},\quad l_\beta(k):= \sup_{u\in Z_k, \|u\|=1 } \|u\|_{L^\beta}.$$ Since $E$ is compactly embedded into $L^1$ and $L^\beta$ respectively, $$l_1(k)\to 0,\quad l_\beta(k)\to 0,\quad \text{as } k\to \infty.\label{e3.16}$$ Consequently, for any $k\ge \bar{n}+1$, \eqref{e3.15} implies $$\Phi_\lambda (u)\ge \frac{1}{2} \Big( 1 -\delta^2|A|\Big)\|u\|^2 -2c_1\Big(l^\beta_\beta(k)\|u\|^{\beta }+ l_1(k)\|u\|\Big), \label{e3.17}$$ for all $(\lambda, u)\in [1,2]\times Z_k$. Let $$\rho_k=\frac{8c_1}{1-\delta^2|A|} \Big(l^\beta_\beta(k)+ l_1(k)\Big), \quad \forall k\in \mathbb{N}.$$ From \eqref{e3.16}, we obtain $$\rho_k\to 0\quad \text{as } k\to \infty,\label{e3.18}$$ and there exists $k_0> \bar{n}+1$ such that $$\rho_k<1,\quad \forall k\ge k_0.\label{e3.19}$$ Combining \eqref{e3.17}-\eqref{e3.19} and the definition of $\rho_k$, a straightforward computation shows that $$a_k(\lambda):=\inf_{u\in Z_k, \|u\|=\rho_k }\Phi_\lambda (u) \ge \frac{1-\delta^2|A|}{4}\rho_k^2> 0,\quad \forall k\ge k_0.$$ Furthermore, by \eqref{e3.17}, for any $k \ge k_0$ and $u\in Z_k$ with $\|u\|\le \rho_k$, we have $$\Phi_\lambda (u)\ge -2c_1\Big(l^\beta_\gamma(k)\rho_k^{\beta }+ l_1(k)\rho_k\Big).$$ Then $$0\ge \inf_{u\in Z_k, \|u\|\le\rho_k }\Phi_\lambda (u) \ge -2c_1\Big(l^\beta_\beta(k)\rho_k^{\beta }+ l_1(k)\rho_k\Big), \quad \forall k\ge k_0.$$ Combining \eqref{e3.16} and \eqref{e3.18}, we obtain $$d_k(\lambda):=\inf_{u\in Z_k, \|u\|\le\rho_k }\Phi_\lambda (u)\to 0\quad \text{as k\to\infty, uniformly for } \lambda \in [1,2].$$ \end{proof} \begin{lemma} \label{lem3.4} Under the assumptions of Theorem \ref{thm1.1}, there exists $00$ depending on $Y_k$. Now, choosing $$0< r_k< \min \{ \rho_k, \frac{\omega}{C_\infty}, \delta_k^{1/(2-\nu)}\},\quad \forall k\in\mathbb{N}.$$ By \eqref{e3.20}, a direct computation gives $$b_k(\lambda):=\max_{u\in Y_k, \|u\|=r_k} \Phi_\lambda (u)\le \frac{\delta^2|A|-1}{2}r_k^2<0,\quad \forall k\in\mathbb{N}.$$ \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] Combining Lemma \ref{lem2.1}, lemma \ref{lem2.2}, \eqref{e3.1} and \eqref{e3.14}, it is easy to see that $\Phi_\lambda$ maps bounded sets to bounded sets uniformly for $\lambda \in [1,2]$. Moreover, by (V4), $\Phi_\lambda (-u)=\Phi_\lambda (u)$ for all $(\lambda,u) \in [1,2]\times E$. Thus the condition (T1) of Theorem \ref{thm2.5} holds. Lemma \ref{lem3.2} shows that the condition (T2) holds, while Lemma \ref{lem3.3} together with Lemma \ref{lem3.4} imply that the condition (T3) holds for all $k\ge k_0$, where $k_0$ is given in Lemma \ref{lem3.3}. Therefore, by Theorem \ref{thm2.5}, for each $k\ge k_0$, there exist $\lambda_n \to 1, u_{\lambda_n} \in Y_n$ such that $$\Phi'_{\lambda_n} |_{Y_n}(u_{\lambda_n})=0, \quad \Phi_{\lambda_n} (u_{\lambda_n})\to f_k \in [d_k(2), b_k(1)]\quad \text{as } n\to \infty.\label{e3.21}$$ It remains to prove that the sequence $\{u_{\lambda_n}\}$ is bounded. Otherwise, we suppose, up to a subsequence, that $$\|u_{\lambda_n}\|\to \infty, \quad \text{as} \ n\to\infty.\label{e3.22}$$ Let $u_n:=u_{\lambda_n}=u_n^- +u_n^0 +u_n^+$ in $E = E^- \oplus E^0 \oplus E^+$ and assume that $$u_n/\|u_n\|\rightharpoonup w,\quad u_n^\pm/\|u_n\|\rightharpoonup w^\pm, \quad u_n^0/\|u_n\|\rightharpoonup w^0.$$ By \eqref{e3.21}, we have $$(u_n^+,v_n)-\lambda_n(u_n^-,v_n) +\int_\mathbb{R} (Au_n(t),\dot{v}_n(t))dt -\lambda_n\int_\mathbb{R}(\nabla V(t,u_n(t)),v_n(t)) dt=0,\label{e3.23}$$ where $v_n=v|_{Y_n}$, $v=\sum_{i=1}^\infty s_i e_i$. Using (V3) and Lemma \ref{lem2.2} we can find a constant $d>0$ such that \begin{aligned} \big|\int_\mathbb{R}(\nabla V(t,u_n(t)),v_n(t)) dt\big| &\le a_4\int_\mathbb{R} a(t)|u_n(t)|^{\beta -1}|v_n(t)|dt + a_4\int_\mathbb{R} a(t)|v_n(t)|dt\\ & \le d\Big( \|u_n\|^{\beta -1} + 1\Big)\|v_n\| \end{aligned} \label{e3.24} Since ${\beta -1}<1$, from \eqref{e3.22} and \eqref{e3.24}, we obtain $$\frac{1}{\|u_n\|}\int_\mathbb{R}(\nabla V(t,u_n(t)),v_n(t)) dt\to 0,\quad \text{as } n\to\infty.\label{e3.25}$$ Also, dividing by $\|u_n\|$ in \eqref{e3.23} and passing to the limit, we obtain $$(w^+,v)-(w^-,v) +\int_\mathbb{R} (Aw(t),\dot{v}(t))dt =0.\label{e3.26}$$ If $w\neq 0$, \eqref{e3.26} is equivalent to $0\in \sigma\Big(-(d^2/dt^2)+ L(t) - A (d/dt)\Big)$ which contradicts assumption (L3). If $w=0$. From \eqref{e3.21}, we have \begin{aligned} 0&=\lambda_n\Big(\|u_n\|^2-\|u_n^0\|^2\Big) +\int_\mathbb{R} (Au_n,\lambda_n\dot{u}_n^+ - \dot{u}_n^-)dt\\ &\quad -\lambda_n\int_\mathbb{R}(\nabla V(t,u_n),\lambda_n u_n^+ - u_n^-) dt. \end{aligned} \label{e3.27} Arguing as in \eqref{e3.24}-\eqref{e3.25}, we obtain $$\frac{1}{\|u_n\|^2}\int_\mathbb{R}(\nabla V(t,u_n),\lambda_n u_n^+ - u_n^-) dt \to 0\quad \text{as } n\to\infty.\label{e3.28}$$ Combining \eqref{e3.22}, \eqref{e3.27} and \eqref{e3.28} we obtain $$\frac{1}{\|u_n\|^2}\int_\mathbb{R} (Au_n,\lambda_n\dot{u}_n^+ - \dot{u}_n^-)dt\to -1\quad \text{as } n\to\infty.\label{e3.29}$$ On the other hand, by Lemma \ref{lem2.2}, passing if necessary to a subsequence, we have $\frac{u_n}{ \|u_n\|}\to 0$ in $L^2$. Also, by Lemma \ref{lem2.1}, the sequence $\{\frac{\lambda_n\dot{u}_n^+ - \dot{u}_n^-}{\|u_n\|}\}$ is bounded in $L^2$, so it is obvious that $$\frac{1}{\|u_n\|^2}\int_\mathbb{R} (Au_n,\lambda_n\dot{u}_n^+ - \dot{u}_n^-)dt\to 0\quad \text{as}\ n\to\infty.$$ This is in contradiction with \eqref{e3.29}. Therefore, $\{u_{n}\}$ is bounded and by a standard argument it possesses a strong convergent subsequence in $E$ (see \cite{z1,z4}). Now, from the last assertion of Theorem \ref{thm2.5}, we know that $\Phi=\Phi_1$ has infinitely many nontrivial critical points and by Lemma \ref{lem3.1}, system \eqref{DS} possesses infinitely many nontrivial homoclinic solutions. This completes the proof. \end{proof} \section{Proof of Theorem \ref{thm1.4}} The proof is based on the following two lemmas. \begin{lemma} \label{lem4.1} Under the conditions of Theorem \ref{thm1.4}, $\Phi\in C^1(E,\mathbb{R})$ and \begin{align*} \Phi'(u)v&=\int_\mathbb{R} \Big[(\dot u(t),\dot v(t))+(L(t)u(t),v(t))\Big]dt \\ &\quad +\int_\mathbb{R} (Au(t),\dot{v}(t))dt -\int_\mathbb{R}(\nabla V(t,u(t)),v(t)) dt. \end{align*} for all\ $u = u^- + u^0 +u^+$, $v = v^- + v^0 +v^+$ in $E = E^- \oplus E^0 \oplus E^+$. Moreover, any critical point of $\Phi$ on $E$ is a homoclinic solution of \eqref{DS}. \end{lemma} \begin{proof} Using the notation of Lemma \ref{lem3.1}, we need to prove that $\Psi_3\in C^1(E,\mathbb{R})$ and $$\Psi_3'(q)v=\int_\mathbb{R} (\nabla V(t,q(t)),v(t))dt,\quad \forall q,v \in E.$$ Let $u \in E$, from Lemma \ref{lem2.1}, we know that $u \in W^{1,2}(\mathbb{R}, \mathbb{R}^N)$ and hence there exists $T_0> 0$ such that $$| u(t)| \leq r/2, \quad \forall | t|\geq T_0.\label{e4.1}$$ By \eqref{e2.3}, for any $v\in E$ with $\|v\|\le \frac{r}{2C_\infty}$, we have $$\|v\|_{L^\infty}\le r/2. \label{e4.2}$$ Combining \eqref{e4.1}, \eqref{e4.2} and (V4'), by the mean value theorem and the H\"{o}lder inequality, for any $T > T_0$ and $v\in E$ with $\|v\|\le \frac{r}{2C_\infty}$, we have \begin{aligned} &\Big|\int_{|t|> T} \Big[ V(t, u+v)- V(t,u)- (\nabla V(t,u),v)\Big] dt\Big|\\ &=\Big|\int_{|t|> T} \Big[\int_0^1 (\nabla V(t,u+sv) -\nabla V(t,u),v)ds\Big] dt\Big|\\ &\le2a_2 \int_{|t|> T} (|u|+ |v|)^\beta |v| dt\\ &\le2a_2 \Big(\int_{|t|> T} (|u|+ |v|)dt\Big)^\beta \|v\|_{L^{\frac{1}{1-\beta}}}\\ & \le 2a_2 C_{\frac{1}{{1-\beta}}} \Big(\int_{|t|> T} (|u|+ |v|)dt\Big)^\beta \|v\|. \end{aligned} \label{e4.3} In view of Lemma \ref{lem2.2}, for any $\varepsilon >0$, there exist $0<\delta_1\le \frac{r}{2C_\infty}$ and $T_\varepsilon > T_0$ such that $$2a_2 C_{\frac{1}{{1-\beta}}}\Big(\int_{|t|> T_\varepsilon} (|u|+ |v|)dt\Big)^\beta \le \varepsilon/2, \quad \forall v\in E,\; \|v\|\le \delta_1.\label{e4.4}$$ Define $\Psi_T : W^{1,2}([-T,T], \mathbb{R}^N)\to \mathbb{R}$ by $$\Psi_T (u)=\int_{-T}^T V(t,u) dt, \quad \forall u\in W^{1,2}([-T,T], \mathbb{R}^N).$$ It is known (see, e.g., \cite{r3}) that $\Psi_T\in C^1( W^{1,2}([-T,T], \mathbb{R}^N))$ for any $T>0$. Combining this with the fact $E$ is continuously embedded in $W^{1,2}(\mathbb{R}, \mathbb{R}^N)$ from Lemma \ref{lem2.1}, for the $\varepsilon$ and $T_\varepsilon$ given above, there exists $\delta_2=\delta_2(u,\varepsilon ,T_\varepsilon)$ such that $$\big|\int_{-T_\varepsilon}^{T_\varepsilon} \big[ V(t, u+v)- V(t,u)- (\nabla V(t,u),v)\big] dt\big| \le \frac{\varepsilon}{2}\|v\|, \quad \forall v\in E, \; \|v\|\le \delta_2.\label{e4.5}$$ Combining \eqref{e4.3}-\eqref{e4.5} and taking $\delta= \min\{ \delta_1, \delta_2\}$, we obtain $$\big|\int_\mathbb{R} \big[ V(t, u+v)- V(t,u)- (\nabla V(t,u),v)\big] dt\big| \le \varepsilon\|v\|, \quad \forall v\in E, \; \|v\|\le \delta.$$ Thus $\Psi_3$ is Fr\'echet differentiable and $$\Psi_3'(q)v=\int_\mathbb{R} (\nabla V(t,q(t)),v(t))dt,\quad \forall q,v \in E.$$ Next we prove that $\Psi'_3$ is weakly continuous. Let $u_n\rightharpoonup u_0$ in $E$. Again, using Lemma \ref{lem2.2}, $u_n \to u_0$ in $L^p$ for all $1\le p\le \infty$. By the H\"{o}lder inequality, \begin{aligned} \|\Psi_3'(u_n)-\Psi_3'(u_0)\|_{E^*} &=\sup_{\|v\|=1} \|(\Psi_3'(u_n)-\Psi_3'(u_0))v\| \\ &= \sup_{\|v\|=1}\Big|\int_\mathbb{R} (\nabla V(t, u_n)-\nabla V(t, u_0), v) dt\Big|\\ &\le \sup_{\|v\|=1}\Big[\Big(\int_\mathbb{R} |\nabla V(t, u_n) -\nabla V(t, u_0)|^3 dt\Big)^{1/3} \|v\|_{3/2}\Big]\\ &\le C_{3/2}\Big(\int_\mathbb{R} |\nabla V(t, u_n) -\nabla V(t, u_0)|^3 dt\Big)^{1/3},\quad \forall n\in \mathbb{N}, \end{aligned} \label{e4.6} Since $u_n\to u_0$ in $L^1$, there exists a constant $M_0> 0$ such that $$\|u_n\|_{L^1}\le M_0,\quad \forall n\in \mathbb{N}.\label{e4.7}$$ By (V4'), for any $\varepsilon >0$, there exists $\eta >0$ such that $$|\nabla V(t,u)|\le \frac{\varepsilon}{2(M_0^{1/3} +\|u_0\|_{L^1}^{1/3})}|u|^{1/3}, \quad \forall u\in \mathbb{R}, |u|\le \eta. \label{e4.8}$$ Due to \eqref{e4.8}, the fact that $u_0\in W^{1,2}(\mathbb{R}, \mathbb{R}^N)$ and $u_n\to u_0$ in $L^\infty$, there exist $T_\varepsilon'>0$ and $N_1\in \mathbb{N}$ such that for all $n> N_1$ and $|t|\ge T_\varepsilon'$, $$\begin{gathered} |\nabla V(t,u_n)|\le \frac{\varepsilon}{2(M_0^{1/3} +\|u_0\|_{L^1}^{1/3})}|u_n|^{1/3}, \\ |\nabla V(t,u_0)|\le \frac{\varepsilon}{2(M_0^{1/3} +\|u_0\|_{L^1}^{1/3})}|u_0|^{1/3}. \end{gathered}\label{e4.9}$$ By \eqref{e4.7} and \eqref{e4.9}, we have \begin{aligned} &\Big(\int_{|t|\ge T_\varepsilon'} |\nabla V(t, u_n) -\nabla V(t, u_0)|^3 dt\Big)^{1/3}\\ &\le \frac{\varepsilon}{2(M_0^{1/3} +\|u_0\|_{L^1}^{1/3})} (\|u_n\|_{L^1}^{1/3}+\|u_0\|_{L^1}^{1/3})\\ & \le \frac{\varepsilon}{2},\quad \forall n\ge \mathbb{N}. \end{aligned} \label{e4.10} On the other hand, using $u_n\to u_0$ in $L^\infty$ and (V3'), by Lebesgue's Dominated Convergence Theorem, $$\Big(\int_{- T_\varepsilon'}^{T_\varepsilon'} |\nabla V(t, u_n)-\nabla V(t, u_0)|^3 dt\Big)^{1/3} \to 0\quad \text{as } n\to\infty.$$ Thus there exists $N_2\in \mathbb{N}$ such that for all $n> N_2$, $$\Big(\int_{- T_\varepsilon'}^{T_\varepsilon'} |\nabla V(t, u_n)-\nabla V(t, u_0)|^3 dt\Big)^{1/3}\le \varepsilon/2.$$ Combining the last inequality with \eqref{e4.10} and taking $N_\varepsilon= \max\{N_1, N_2\}$, we obtain $$\Big(\int_\mathbb{R} |\nabla V(t, u_n)-\nabla V(t, u_0)|^3 dt\Big)^{1/3} \le \varepsilon, \quad \forall n\ge N_\varepsilon.\label{e4.11}$$ Inequality \eqref{e4.11} with \eqref{e4.6} imply the continuity of $\Psi_3'$ and therefore $\Psi_3\in C^1(E,\mathbb{R})$. The rest of the proof is similar to that of Lemma \ref{lem3.1}. \end{proof} \begin{lemma} \label{lem4.2} Under the assumption {\rm (V1')}, $\mathcal{B}(u)\ge 0$ and $\mathcal{B}(u)\to\infty$ as $\|u\| \to \infty$ on any finite dimensional subspace of $E$. \end{lemma} \begin{proof} Evidently, $\mathcal{B}(u)\ge 0$. An argument similar to but easier than the proof of \eqref{e3.6} allows to claim that for any finite dimensional subspace $F\subset E$, there exists $\epsilon >0$ such that $$\operatorname{meas}(\{t\in \mathbb{R} ; |u(t)|\ge \epsilon \|u\|\}) \ge\epsilon, \quad \forall u \in F\backslash\{0\}.\label{e4.12}$$ By (V1'), for any $A> 0$, there exists $B>0$ such that $$V(t, x) \ge A/\epsilon, \quad \forall \ t\in \mathbb{R} \text{ and } |x|\ge B.\label{e4.13}$$ where $\epsilon$ is given in \eqref{e4.12}. Let $$\Lambda_u=\{t\in \mathbb{R} : |u(t)|\ge \epsilon \|u\|\},\quad \forall u \in F\backslash\{0\}.$$ Then by \eqref{e4.12}, $$\operatorname{meas}(\Lambda_u)\ge \epsilon,\quad \forall u \in F\backslash\{0\}. \label{e4.14}$$ Observing that for $u\in F$ with $\|u\|\ge B/\epsilon$, there holds $$|u(t)|\ge B,\quad \forall \ t\in\Lambda_u. \label{e4.15}$$ Combining \eqref{e4.13}-\eqref{e4.15}, for any $u\in F$ with $\|u\|\ge B/\epsilon$, we have \begin{align*} \mathcal{B}(u) &=\frac{1}{2} \|u^-\|^2 +\int_\mathbb{R} V(t,u(t))dt\\ &\ge\int_{\Lambda_u} V(t,u(t))dt\\ &\ge \operatorname{meas}(\Lambda_u) A/\epsilon \ge A , \end{align*} which implies that $\mathcal{B}(u)\to\infty$ as $\|u\| \to \infty$ on $F$. \end{proof} To complete the proof of Theorem \ref{thm1.4}, we observe that since (V3') is the particular case of (V3) where $\beta=1$, then Lemma \ref{lem3.3} remains true under the assumption (V3'). Also, it is obvious that Lemma \ref{lem3.4} still holds with (V2') replacing (V2). 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