\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 134, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2006/134\hfil Periodic solutions] {Periodic solutions for second-order Hamiltonian systems with the p-Laplacian} \author[Y. Tian, W. Ge\hfil EJDE-2006/134\hfilneg] {Yu Tian, Weigao Ge} \address{Yu Tian \newline School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China} \email{tianyu2992@163.com} \address{Weigao Ge \newline Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China} \email{gew@bit.edu.cn} \date{} \thanks{Submitted July 16, 2006. Published October 19, 2006.} \thanks{Supported by grants 10671012 from the National Natural Sciences Foundation of China and \hfill\break\indent 20050007011 from Foundation for Ph.D. Specialities of Educational Department of China.} \subjclass[2000]{34B15, 58E30} \keywords{ Hamiltonian system; p-Laplacian; Mountain Pass Theorem; \hfill\break\indent Periodic solution} \begin{abstract} In this paper, we investigate the periodic solutions of Hamiltonian system with the p-Laplacian. By using Mountain Pass Theorem the existence of at least one periodic solution is obtained, Furthermore, under suitable assumptions, we obtain the existence of infinitely many solutions via $Z_2$-symmetric version of the Mountain Pass Theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} There has been published an extensive literature related to the existence of the periodic solutions of second-order differential equations (systems) recently; see for example the refeences in this article and the references cited therein. In \cite{Bahri,benci} the authors considered the system $$\begin{gathered} \ddot{u}(t)+\nabla H(t, u(t))=0\quad \text{a.e. } t\in[0, T],\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0, \end{gathered}\label{01}$$ and obtained multiple solutions under the following assumption on the potential $H$: There exist $R>0$, $\theta\in]0, 1/2[$ such that 02$,$b\in C([0, T], R)$satisfying$\int_{0}^{T}b(t)dt=0$,$b\not\equiv0$and$H: [0, T] \times \mathbb{R}^N\to R$,$H(t, x)$is measurable in$t$for every$x\in \mathbb{R}^N$and continuously differentiable in$x$for a.e.$t\in[0, T]$, such that \begin{itemize} \item[(A1)]$\int_{0}^{T}H(t, x)dt\ge 0$for all$x\in \mathbb{R}^N$. \item[(A2)] There exist$g\in L^{1}(0, T)$,$\alpha_0\in (0, \omega^2/2)$and$r_0>0$such that$|\nabla H(t, x)|\le g(t)$for all$x\in \mathbb{R}^N$and a.e.$t\in[0, T]$. \item[(A3)]$|H(t, x)|\le \alpha_0|x|^2$for all$|x|\le r_0$and a.e.$t\in[0, T]$, where$\omega=2\pi/T$. \end{itemize} Then the problem \begin{gather*} \ddot{u}(t)+b(t)|u(t)|^{\mu-2}u(t)+\nabla H(t, u(t))=0\quad \text{a.e. }t \in [0, T],\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0, \end{gather*} has at least one nonzero solution. \end{theorem} However, in [1-14, 16, 17], the highest order derivatives are linear. But so far, few papers discuss periodic solutions for second order system with the p-Laplacian. On the other hand, it is well known that the study of the existence for quasilinear differential equations is very important. Motivated by the above works, we consider the existence of solutions for the following second-order Hamiltonian system with p-Laplacian: $$\label{000} \begin{gathered} \frac{d}{dt}(\Phi_p(\dot{u}(t)))+B(t)\Phi_{\mu}(u(t)) +\nabla H(t, u(t))=0,\quad t\in[0, T],\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0, \end{gathered}$$ where$p>1$,$\Phi_p(u):=(|u_1|^{p-2}u_1, \dots, |u_N|^{p-2}u_N)$,$u=(u_1, \dots, u_N)$,$\mu>p$,$T>0$,$H: [0,T]\times \mathbb{R}^N\to R$,$H(t, x)$is measurable in$t$for every$x\in \mathbb{R}^N$and continuously differentiable in$x$for a.e.$t\in[0, T]$,$\nabla H(t, x)=(\partial H/\partial x)(t, x)$. $B(t)=\begin{bmatrix} b_1(t)\\ & &\ddots\\ & & & b_N(t) \end{bmatrix},$$b_i\in C([0, T], R)$,$b_i\not\equiv 0$,$i=1, 2,\dots, N$. For$p=2$,$\frac{d}{dt}(\Phi_p(\dot{u}(t)))=\ddot{u}(t)$. To apply critical point theory to \eqref{1}, it is necessary to check that corresponding functional verifies the Palais-Smale condition ((PS)-condition). Taking the quasi-linear into consideration, uniformly convex of$L^p$and related differential inequalities have to be used (see section 2). Using the Mountain Pass Theorem, the existence of at least one solution is obtained. Furthermore, under the hypothesis of eveness of the functional, the existence of infinitely many solutions is obtained by using$Z_2$version of the Mountain Pass Theorem. The following lemmas are taken from \cite{zhuanzhu}, and will be useful in the proofs of our main results. \begin{lemma}\label{th3} Let$E$be a real Banach space with$E=V\oplus X$, where$V$is finite dimensional. Suppose$I\in C^{1}(E, R)$satisfies (PS), and \begin{itemize} \item[(I1)] There are constants$\rho, \alpha>0$such that$I_{\partial B_{\rho}\cap X}\ge \alpha$, and \item[(I2)] There is an$e\in \partial B_1\cap X$and$R>\rho$such that if$Q\equiv (\overline{B}_R\cap V)\oplus \{re:00$such that$I|_{\partial B_{\rho}\cap X}\ge \alpha$and \item[(J2)] for each finite dimensional subspace$V_1\subset E$, the set$\{x\in V_1: I(x)\ge 0\}$is bounded. \end{itemize} Then$I$has an unbounded sequence of critical values. \end{lemma} \section{Preliminaries} In the proof of main results, we will need the following preliminary results. For convenience, let $W_{T}^{1, p}([0, T])=\big\{u: [0, T]\mapsto \mathbb{R}^N: u \mbox{ is abs. cont., } u(0)=u(T),\; \dot{u}\in L^p(0, T; \mathbb{R}^N)\big\}$ be a Sobolev space with the norm $\|u\|_{W_{T}^{1, p}}=\Big(\int_{0}^{T}|u(t)|^p+|\dot{u}(t)|^pdt\Big)^{1/p}.$ For any$x\in W_{T}^{1, p}([0,T])$, define$A: W_{T}^{1, p}([0, T])\to R$by $$\label{A} A(x)=\int_{0}^{T}\sum_{i=1}^{N}|\dot{x}_{i}(t)|^{p}dt =\sum_{i=1}^{N}\|\dot{x}_{i}\|_{L^p}^{p}.$$ Let$A_i(x)=\|\dot{x}_{i}\|_{L^p}^{p}$, so$A(x)=\sum_{i=1}^{N}A_i(x)$. Clearly,$A$is convex. Now we claim that$A$is lower semi-continuous on$W_{T}^{1, p}([0, T])$. So$A$is weakly lower semi-continuous on$W_{T}^{1, p}([0, T])$(see \cite[Theorem 1.2]{mawhin}). In fact, we only need to show$A_i$,$i\in\{1, 2, \dots, N\}$is weakly lower semi-continuous on$W_{T}^{1, p}([0, T])$. Let$x_n\to x$in$W_{T}^{1, p}([0,T])$, from which it follows$\|\dot{x}_n-\dot{x}\|_{L^p}\to0$. By $\|x+y\|_{L^p}\le \|x\|_{L^p}+\|y\|_{L^p}\quad \forall x, y\in L^{p}([0, T]),$ we have $A_i(x_n)-A_i(x)=\|\dot{x}_{ni}\|_{L^p}^p-\|\dot{x}_{i}\|_{L^p}^{p}\le \left(\|\dot{x}_{ni}-\dot{x}_i\|_{L^p}+\|\dot{x}_i\|_{L^p}\right)^{p} -\|\dot{x}_i\|_{L^p}^{p}\to 0$ as$n\to\infty, and \begin{align*} A_i(x_n)-A_i(x) &=\|\dot{x}_{ni}\|_{L^p}^p-\|\dot{x}_{i}\|_{L^p}^{p}\\ &\ge \Big(\Big|\|\frac{\dot{x}_{ni}+\dot{x}_i}{2}\|_{L^p} -\|\frac{\dot{x}_{ni}-\dot{x}_i}{2}\|_{L^p}\Big|\Big)^p\\ & \quad - \left(\|\frac{\dot{x}_{ni}+\dot{x}_i}{2}\|_{L^p} -\|\frac{-\dot{x}_{ni}+\dot{x}_i}{2}\|_{L^p}\right)^p\to 0\quad \mbox{as } n\to\infty. \end{align*} Therefore, $\lim_{n\to\infty}A_i(x_n)=A_i(x).$ \begin{lemma}[\cite{uniform}] \label{le1} For the spaceL^{p}([0, T])$, the following inequalities between the norms of two arbitrary elements$x$and$y$of the space are valid (here$q$is the conjugate index,$q=p/(p-1)$): \begin{gather}\label{ineq1} \|\frac{x+y}{2}\|_{L^p}^{p}+\|\frac{x-y}{2}\|_{L^p}^{p}\le \frac{1}{2}(\|x\|_{L^{p}}^{p}+\|y\|_{L^p}^{p})\quad \mbox{for }p\ge2,\\ \label{ineq2} \|\frac{x+y}{2}\|_{L^p}^{q}+\|\frac{x-y}{2}\|_{L^p}^{q}\le \big[\frac{1}{2}(\|x\|_{L^{p}}^{p}+\|y\|_{L^p}^{p})\big]^{q-1}\quad \mbox{for }10$ and subsequence $(x_{nk})$ that satisfy $\|x_{nk}-x\|_{W_{T}^{1, p}}\ge \varepsilon$. Since $(x_{n_k})$ converges uniformly to $x$ in $C([0, T])$, $\|\dot{x}_{nk}-\dot{x}\|_{L^p}\ge \varepsilon$. So there exists $j\in\{1,2,\dots, N\}$ and $\widetilde{\varepsilon}>0$, $\|\dot{x}_{nkj}-\dot{x}_j\|_{L^p}\ge \widetilde{\varepsilon}$. From \eqref{ineq1}, \begin{align*} \limsup_{n_k\to\infty}A\big(\frac{x+x_{nk}}{2}\big) &\le \limsup_{n_k\to\infty} \frac{1}{2}A(x)+\frac{1}{2}A(x_{nk})-A\big(\frac{x-x_{nk}}{2}\big) \\ &\le c- \limsup_{n_k\to\infty}\sum_{i=1}^{N}\|\frac{\dot{x}_{nki} -\dot{x}_i}{2}\|_{L^p}^{p} \\ &\le c-\frac{1}{2^p}\widetilde{\varepsilon}^{p}, \end{align*} which contradicts \eqref{**}. For $10 \] such that $$\label{02}| \nabla H(t, x)|\le g(t)|x|^{\theta}$$ for all$x\in \mathbb{R}^N$and a.e.$t\in[0, T]$, and $$\label{1.5'} |H(t, x)|\le h(t)|x|^{p}$$ for$|x|\le r$and a.e.$t\in[0, T]$. Then system \eqref{000} has at least one nonzero solution. \end{theorem} \begin{proof} The proof is divided into three steps. \noindent {\it Step 1.} We claim that the functional$\varphi$satisfies the Palais-Smale condition, that is,$(u_n)$has a convergent subsequence whenever it satisfies$\varphi'(u_n)\to 0$as$n\to \infty$and$\{\varphi(u_n)\}$is bounded. First we prove that$(u_n)$is a bounded sequence in$W_{T}^{1, p}([0, T])$. Suppose that$(u_n)$is unbounded. Passing to a subsequence, we may assume if necessary, that$\|u_n\|_{W_{T}^{1, p}}\to\infty$as$n\to\infty$. Set$v_n=\frac{u_n}{\|u_n\|_{W_{T}^{1, p}}}$. Then$(v_n)$is bounded so that it has a subsequence, say$(v_n)$, which weakly converges to$v_0$. By \cite[Proposition 1.2]{mawhin},$(v_n)$converges to$v_0$uniformly on$[0, T]$. Hence one has$\overline{v}_n\to\overline{v}_0$, where$\overline{v}=\frac{1}{T}\int_{0}^{T}v(s)ds. It follows from \eqref{*} and \eqref{02} that \label{*)} \begin{aligned} &\mu\int_{0}^{T}H(t, u_n(t))dt-\int_{0}^{T}(\nabla H(t, u_n(t)), u_n(t))dt\\ &\le \mu\int_{0}^{T}\left(\int_{0}^{1} \left(\nabla H(t, su_n(t)), u_n(t)\right)ds\right)dt +\mu\int_{0}^{T}H(t, 0)dt \\ &\quad +\int_{0}^{T}|\nabla H(t, u_n(t))||u_n(t)|dt\\ &\le\frac{\mu}{\theta+1}\int_{0}^{T}g(t)|u_n(t)|^{\theta+1}dt +\mu\int_{0}^{T}H(t,0)dt+\int_{0}^{T}g(t)|u_n(t)|^{\theta+1}dt\\ &\le \big(\frac{\mu}{\theta+1}+1\big)\|g\|_{L^1}C^{\theta+1}\|u_n\|_{W_{T}^{1, p}}^{\theta+1}+\mu\int_{0}^{T}H(t,0)dt. \end{aligned} Then by \eqref{3.0}, \eqref{04}, \eqref{*)}, we have \begin{align*} &\big(\frac{\mu}{p}-1\big)\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_{ni}(t)|^{p}dt\\ &=\mu \varphi(u_n)-\langle\varphi'(u_n), u_n\rangle +\int_{0}^{T}\mu H(t, u_n)dt -\int_{0}^{T}(\nabla H(t, u_n), u_n)dt\\ &\le \mu \varphi(u_n)-\langle\varphi'(u_n), u_n\rangle+\big(\frac{\mu}{\theta+1}+1\big)\|g\|_{L^1}C^{\theta+1} \|u_n\|_{W_{T}^{1, p}}^{\theta+1}+\mu\int_{0}^{T}H(t, 0)dt, \end{align*} which implies\|\dot{v}_{ni}\|_{L^p}\to 0$as$n\to\infty$,$i=1, 2, \dots, N$. So$\widetilde{v}_n\to 0$in$W_{T}^{1, p}([0, T])$as$n\to\infty$, thus$v_n\to \overline{v}_0$as$n\to\infty$, where$\widetilde{v}=v-\overline{v}$. Hence$v_0=\overline{v}_0$and$\|v_0\|_{W_{T}^{1, p}}=1. On the other hand, from \eqref{04}, \begin{align*} &\Big|\int_{0}^{T}(B(t)\Phi_{\mu}(v_n(t)), v(t))dt\Big| \\ &=\Big|\int_{0}^{T}\sum_{i=1}^{N}b_i(t)|v_{ni}(t)|^{\mu-2}v_{ni}(t)v_i(t)dt \Big|\\ &\le \|u_n\|_{W_{T}^{1, p}}^{1-\mu}|\langle\varphi'(u_n), v\rangle|+\|u_n\|_{W_{T}^{1, p}}^{p-\mu}\Big|\int_{0}^{T}\sum_{i=1}^{N}|\dot{v}_{ni}(t)|^{p-2} \dot{v}_{ni}(t)\dot{v}_i(t)dt\Big| \\ &\quad +\|u_n\|_{W_{T}^{1, p}}^{1-\mu} \Big|\int_{0}^{T}(\nabla H(t, u_n(t)),v(t))dt\Big|\\ &\le \|u_n\|_{W_{T}^{1,p}}^{1-\mu}\|\varphi'(u_n)\|\|v\| +\|u_n\|_{W_{T}^{1,p}}^{p-\mu}\sum_{i=1}^{N}\|\dot{v}_{ni}\|_{L^p}^{p-1} \|\dot{v}_i\|_{L^p}\\ &\quad +C^{\theta}\|u_n\|_{W_{T}^{1,p}}^{1-\mu+\theta} \|g\|_{L^1}\|v\|_{\infty}, \end{align*} asn\to\infty$, which implies that $\Big|\int_{0}^{T}(B(t)\Phi_{\mu}(v_0(t)), v(t))dt\Big|=0$ for all$v\in W_{T}^{1, p}([0, T])$. By the arbitrariness of$v$, one has $B(t)\Phi_{\mu}(v_0(t))=0$ for a.e.$t\in[0, T]$. Because$v_0=\overline{v}_0\neq 0$, we have$b_i(t)=0$for a.e.$t\in[0, T]$. It follows from the continuity of$b_i$that$b_i=0$for$t\in[0, T]$, which contradicts the condition$b_i\not\equiv 0$. Hence$(u_n)$is a bounded sequence. From the reflexivity of$W_{T}^{1, p}([0, T])$, we extract a weakly convergent subsequence, that for simplicity, we call$(u_n), u_n\rightharpoonup u$. Following we will show that$(u_n)$converges strongly to$u$. To this end, we note that$A(u)=\sum_{i=1}^{N}\int_{0}^{T}|\dot{u}_i(t)|^{p}dt. From Proposition \ref{prop1} it is enough to prove $\limsup_{n\to\infty}\int_{0}^{T}(DA(u_n), u_n-u)dt =\limsup_{n\to\infty}\int_{0}^{T}\sum_{i=1}^{N} \Phi_p(\dot{u}_{ni}(t))(\dot{u}_{ni}(t)-\dot{u}_{i}(t))dt\le 0.$ From \eqref{04}, it follows \begin{align*} &\int_{0}^{T}\sum_{i=1}^{N}\Phi_p(\dot{u}_{ni}(t))(\dot{u}_{ni}(t) -\dot{u}_i(t))dt\\ &= \langle\varphi'(u_n), u_n-u\rangle +\int_{0}^{T} \sum_{i=1}^{N}b_i(t)\Phi_{\mu}(u_{ni}(t))(u_{ni}(t)-u_i(t))dt\\ &\quad +\int_{0}^{T}(\nabla H(t, u_n(t)), u_n(t)-u(t))dt. \end{align*} Now since the Sobolev embeddingW_{T}^{1, p}([0, T])\hookrightarrow C([0, T])$is compact, we get (for a subsequence) that$u_n\to u$uniformly in$C([0, T])$. Since$\varphi'(u_n)\to 0$by assumption, and$u_n-u$is bounded in$W_{T}^{1, p}([0, T])$, we deduce that$\langle\varphi'(u_n), u_n-u\rangle\to 0. Moreover, \begin{align*} &\int_{0}^{T}\sum_{i=1}^{N} b_i(t)\Phi_{\mu}(u_{ni}(t))(u_{ni}(t)-u_i(t))dt\\ &\le T\sum_{i=1}^{N}\max_{t\in[0, T]} b_i(t)\Phi_{\mu}(\|u_{ni}\|_{\infty})\|u_{ni}-u_i\|_{\infty} \to 0 \end{align*} and \begin{align*} & \int_{0}^{T}(\nabla H(t, u_n), u_n-u)dt\\ &\le \int_{0}^{T}g(t)|u_n|^{\theta}(u_n-u)dt\le\|g\|_{L^1} \|u_n\|_{\infty}^{\theta}\|u_n-u\|_{\infty}\to 0. \end{align*} So\limsup_{n\to\infty}\int_{0}^{T}\sum_{i=1}^{N}\Phi_p(\dot{u}_{ni}) (\dot{u}_{ni}-\dot{u}_i)dt=0$and from Proposition \ref{prop1},$u_n\to u$strongly in$W_{T}^{1, p}([0, T])$. \noindent{\it Step 2.} Let$W_{T}^{1, p}=\mathbb{R}^N\oplus \widetilde{W}_{T}^{1, p}$, where $\widetilde{W}_{T}^{1, p}([0, T])=\{u\in \widetilde{W}_{T}^{1, p}([0, T]): \int_{0}^{T}u(t)dt=0\}.$ We claim there exist$\rho>0, \alpha>0$such that$\varphi(u)\ge \alpha$for all$u\in S=\{u\in \widetilde{W}_{T}^{1, p}([0, T]): \|u\|_{W_{T}^{1, p}}=\rho\}$. By \eqref{1.5'}, for$u\in \widetilde{W}_{T}^{1,p}([0, T]), we have \begin{aligned} \int_{0}^{T}H(t, u)dt &\le \int_{0}^{T}|h(t)|dt\|u\|_{\infty}^{p}=\|h\|_{L^1} \Big(\sum_{i=1}^{N}\|u_i\|_{\infty}^{2}\Big)^{p/2}\\ &\le \|h\|_{L^1}\Big[\sum_{i=1}^{N} \Big(\int_{0}^{T}|\dot{u}_i(t)|dt\Big)^2\Big]^{p/2} \\ &\le \|h\|_{L^1}\max\{N^{\frac{p}{2}-1}, 1\}\sum_{i=1}^{N}\left(\int_{0}^{T}|\dot{u}_i(t)|dt\right)^p\\ &\le T^{\frac{p}{q}} \|h\|_{L^1}\max\{N^{\frac{p}{2}-1}, 1\}\sum_{i=1}^{N}\|\dot{u}_i\|_{L^p}^{p}. \end{aligned}\label{''} From \eqref{1.5'} and \eqref{''} it follows that \begin{align*} \varphi(u)&=\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^{p}dt-\frac{1}{\mu}\int_{0}^{T} \sum_{i=1}^{N}b_i(t)|u_i(t)|^{\mu}dt-\int_{0}^{T}H(t, u(t))dt\\ &\ge \Big(\frac{1}{p}-\|h\|_{L^1}T^{\frac{p}{q}}\max\{N^{\frac{p}{2}-1}, 1\}\Big)\sum_{i=1}^{N}\|\dot{u}_i\|_{L^p}^{p}-\frac{T^{1+\mu/q}}{\mu} \sum_{i=1}^{N}\|b_i\|_{\infty} \|\dot{u}_i\|_{L^p}^{\mu} \end{align*} for allu\in \widetilde{W}_{T}^{1, p}([0, T])$. Since$\mu>p$, we can choose$\rho>0$small enough such that$\varphi(u)\ge \alpha>0$. \noindent {\it Step 3.} By$\int_{0}^{T}b_i(t)dt=0$, there exist$t_{i1}, t_{i2}\in[0, T]$such that$b_i(t)\ge 0$for$t\in[t_{i1}, t_{i2}]$. Choose$e_i\in \widetilde{W}_{T}^{1,p}([0, T])$such that$e_i(t)\equiv 0$for all$t\in[0,t_{i1}]\cup[t_{i2}, T]$,$e_i\not\equiv 0$and$\int_{0}^{T}b_i(t)e_i(t)dt=0, i=1, 2,\dots, N$. Now$e=(e_1, e_2, \dots, e_N)\in \widetilde{W}_{T}^{1, p}([0, T])$. Since$\mathbb{R}^N$is definite dimensional, we will show that there exists$R>\rho>0$such that $\varphi|_{\partial Q}\le 0,\quad Q\equiv\{re:r\in[0, R]\} \oplus (B_{R}\cap \mathbb{R}^N).$ To this end, we denote \begin{gather*} \varphi_1(u)=\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^{p}dt,\quad \varphi_2(u)=-\int_{0}^{T}\sum_{i=1}^{N}b_i(t)|u_i(t)|^{\mu}dt,\\ \varphi_3(u)=-\int_{0}^{T}H(t, u)dt. \end{gather*} Then$\varphi=\varphi_1+\varphi_2+\varphi_3$. By$\sigma\in \mathbb{R}^N$,$\int_{0}^{T}b_i(t)dt=0$and$\int_{0}^{T}b_i(t)e_i(t)dt=0we have \begin{aligned} \int_{t_{i1}}^{t_{i2}}b_i(t)(|\sigma_i|^2+|re_i|^2)dt&= \int_{t_{i1}}^{t_{i2}}b_i(t)|\sigma_i+re_i|^2dt\\ &\le\Big(\int_{t_{i1}}^{t_{i2}}b_i(t)|\sigma_i+re_i|^{\mu}dt\Big)^{2/\mu} \Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\Big)^{\frac{\mu-2}{\mu}}. \end{aligned}\label{3.3} Then one has \begin{aligned} &\varphi_2(\sigma+re)\\ &=-\sum_{i=1}^{N}\int_{t_{i1}}^{t_{i2}}b_i(t)|\sigma_i+re_i(t)|^{\mu}dt +\sum_{i=1}^{N}\int_{t_{i1}}^{t_{i2}}b_i(t)|\sigma_i|^{\mu}dt\\ &\le-\sum_{i=1}^{N}\Big[\int_{t_{i1}}^{t_{i2}}b_i(t)(|\sigma_i|^2 +r^2|e_{i}(t)|^2)dt \Big]^{\mu/2}\Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\Big)^{\frac{2-\mu}{2}}\\ &\quad +\sum_{i=1}^{N}|\sigma_i|^{\mu} \int_{t_{i1}}^{t_{i2}}b_i(t)dt\\ &=-\sum_{i=1}^{N}\Big[\left(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\right)^{2/\mu} |\sigma_i|^2+r^2 \int_{t_{i1}}^{t_{i2}}b_i(t)|e_i(t)|^{2}dt\left(\int_{t_{i1}}^{t_{i2}} b_i(t)dt\right)^{\frac{2-\mu}{\mu}}\Big]^{\mu/2}\\ &\quad +\sum_{i=1}^{N}|\sigma_i|^{\mu}\int_{t_{i1}}^{t_{i2}}b_i(t)dt \end{aligned}\label{3.4} for\sigma\in \mathbb{R}^N, e\in \widetilde{W}_{T}^{1, p}$. Because$\int_{0}^{T}H(t, u)dt>0$for all$u\in \mathbb{R}^N, we have \begin{aligned} \varphi_3(\sigma+re) &= -\int_{0}^{T}H(t, \sigma+re(t))dt\\ &\le -\min\Big\{\int_{0}^{T}H(t,q)dt: q\in \mathbb{R}^N\Big\}\\ &\le 0. \end{aligned} \label{3.5} Therefore, by \eqref{3.4} and \eqref{3.5}, there exist0\rho$large enough such that $$\label{aaa} \varphi(\sigma+re)<0\quad\mbox{ for }\|\sigma\|=R, r\in[0, R], R>R_1.$$ On the other hand, for$\sigma\in \mathbb{R}^N, r=R, by \eqref{3.4}, \begin{aligned} &-\sum_{i=1}^{N}\Big[\ \Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\ \Big)^{2/\mu}|\sigma_i|^2+R^2 \int_{t_{i1}}^{t_{i2}}b_i(t)|e_i(t)|^{2}dt \Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\Big)^{\frac{2-\mu}{\mu}}\Big]^{\mu/2}\\ &+\sum_{i=1}^{N}|\sigma_i|^{\mu}\int_{t_{i1}}^{t_{i2}}b_i(t)dt\\ &\le -R^{\mu}\sum_{i=1}^{N} \Big(\int_{t_{i1}}^{t_{i2}}b_i(t)|e_i(t)|^2dt\Big)^{\mu/2} \Big(\int_{t_{i1}}^{t_{i2}}b_i(t)dt\Big)^{\frac{2-\mu}{2}}. \end{aligned}\label{3.6} Then by \eqref{3.5'} and \eqref{3.6}, \begin{align*} &\varphi(\sigma+Re)\\ &\le \frac{R^p}{p}\sum_{i=1}^{N}\int_{0}^{T}|\dot{e}_i(t)|^{p}dt-R^{\mu}\sum_{i=1}^{N}\left(\int_{t_{i1}}^{t_{i2}}b_i(t)|e_i(t)|^2dt\right)^{\mu/2}\left(\int_{t_{i1}}^{t_{i2}}b_i(t)dt \right)^{\frac{2-\mu}{2}}, \end{align*} which implies that there existsR_2>R_1$such that $$\label{bbb} \varphi(\sigma+Re)\le 0\quad \mbox{for all }\sigma\in \mathbb{R}^N,\; R>R_2.$$ Therefore, \eqref{aaa} and \eqref{bbb} give that$\varphi|_{\partial Q}\le 0$. Hence Theorem \ref{th1} is proved by Lemma \ref{th3}. \end{proof} \begin{remark} \label{rmk3.1} \rm It is clear that our theorem generalizes the results in [1-14, 16, 17] since$p\ge2$. Even for$p=2$, Theorem \ref{th1} generalizes Theorem \ref{thmA} since$\nabla H(t, x)$has more freedom in \eqref{02} than in (A2). There are functions satisfying Theorem \ref{th1} and not satisfying the corresponding results in \cite{Antonacci, btpositive1, chen, Y.H.Ding, btpositive2,tangwu}. For example, let \begin{gather*} b_i(t)=\sin\frac{2\pi t}{T},\quad i=1, 2\dots, N,\\ H(t, u)=\begin{cases} \frac{\pi}{4T^2}\left(\sin\frac{2\pi t}{T}\right)\left(\frac{2}{3}|u|^{\frac{3}{2}}+\sin(|u|-1)+\frac{\pi}{6T}\right), & |u|\ge1,\\ \frac{\pi}{4T^2}\left(\sin\frac{2\pi t}{T}\right)|u|^2,& |u|<1. \end{cases} \end{gather*} A straight forward computation shows that$H(t, u)$satisfies our Theorem \ref{th1} and neither satisfies assumptions in Theorem \ref{thmA}, nor \eqref{1}, hence$H(t, u)$does not satisfy the corresponding results in \cite{Bahri,benci}. Moreover,$F(t, x)=\frac{B(t)}{\mu}|u|^{\mu}+H(t, u)$does not satisfy the conditions of the results in \cite{Antonacci,btpositive1,btpositive2} because that$\int_{0}^{T}b_i(t)dt=0$;$\nabla F(t, x)\to 0$as$|x|\to\infty$, it does not satisfy \cite{chen}. \end{remark} \begin{remark} \label{rmk3.2} \rm For$11$,$H: [0, T]\times \mathbb{R}^N \to R$is even with respect to the second argument and$H(t, 0)=0$. Suppose$b_i\in C(0, T; R^+)$and there exist$g, h\in L^1([0, T])$, with $\|h\|_{L^1}\le [p T^{\frac{p}{q}}\max\{N^{\frac{p}{2}-1}, 1\}]^{-1},\quad \theta\in [0, p-1), \quad r>0$ such that \eqref{02} \eqref{1.5'} hold. Then system \eqref{000} has infinitely many solutions. \end{theorem} \begin{proof} Since$H$is even in the second argument, the functional$\varphi$is even and satisfying$\varphi(0)=0$. By Step 1 in the proof of Theorem \ref{th1},$\varphi$satisfies (PS) condition. Let$W_{T}^{1, p}=\mathbb{R}^N\oplus \widetilde{W}_{T}^{1, p}$. From Step 2 in the proof of Theorem \ref{th1}, there exist$\alpha, \rho>0$such that$\varphi(u)\ge \alpha$if$\|u\|=\rho$,$u\in \widetilde{W}_{T}^{1, p}([0, T]). Now we will verify condition (J2) in Lemma \ref{th3.2}. \begin{aligned} &\varphi(u)\\ &=\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^pddt -\frac{1}{\mu}\int_{0}^{T}\sum_{i=1}^{N}b_i(t) |u_i(t)|^{\mu}dt-\int_{0}^{T}H(t, u(t))dt\\ &\le\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^pdt-\frac{1}{\mu}\int_{0}^{T}\sum_{i=1}^{N}b_i(t) |u_i(t)|^{\mu}dt-\int_{0}^{T} \Big(\int_{0}^{1}(\nabla H(t,su),u)ds\Big)dt\\ &\le\frac{1}{p}\int_{0}^{T}\sum_{i=1}^{N}|\dot{u}_i(t)|^pdt -\frac{1}{\mu}\int_{0}^{T}\sum_{i=1}^{N}b_i(t) |u_i(t)|^{\mu}dt+\int_{0}^{T}\frac{g(t)|u|^{\theta+1}}{\theta+1}dt. \end{aligned} From \eqref{*}, we have \begin{aligned} \|u\|_{\infty}^{\theta+1} &\le \Big(\sum_{i=1}^{N}\|u_i\|_{\infty}^{2}\Big)^{\frac{\theta+1}{2}}\\ &\le C^{\theta+1}\Big(\sum_{i=1}^{N}\|u_i\|_{W_{T}^{1, p}}^{2}\Big)^{\frac{\theta+1}{2}}\\ &\le C^{\theta+1}\max\{N^{\frac{\theta-1}{2}}, 1\}\sum_{i=1}^{N}\|u_i\|_{W_{T}^{1, p}}^{\theta+1}. \end{aligned}\label{3.9'} For any finite dimensional subspaceV_1\subset W_{T}^{1, p}([0, T])$, the norm$\|\cdot\|_{W_{T}^{1,p}}$and$\|\cdot\|_{L^{\theta}}$. So there exists$ \overline{c}>0$such that $$\|u\|_{W_{T}^{1, p}}\le \overline{c}\Big(\int_{0}^{T}|u(t)|^{\mu}dt\Big)^{\frac{1}{\mu}}\quad \mbox{for }u\in V_1.\label{3.12}$$ Moreover by H\"older inequality, there exists a positive constant$\widetilde{c}$such that $$\|u\|_{L^{\theta+1}}\le \widetilde{c}\|u\|_{W_{T}^{1,p}}^{\theta+1}\label{3.13}$$ holds. Thus \eqref{3.9'} \eqref{3.12} and \eqref{3.13} give that there exist$0<\Gamma_i<\infty$,$i=1, 2, 3$such that $\varphi(u)\le \Gamma_1\sum_{i=1}^{N}\|u_i\|_{W_{T}^{1, p}}^{p}-\Gamma_2\sum_{i=1}^{N}\|u_i\|_{W_{T}^{1, p}}^{\mu}+\Gamma_3\sum_{i=1}^{N}\|u_i\|_{W_{T}^{1, p}}^{\theta+1},$ which implies that$\{x\in X_1: \varphi(x)\ge 0\}$is bounded. Then Lemma \ref{th3.2} can be applied to the functional$\varphi$. The proof is completed. \end{proof} \begin{remark} \label{rmk3.3} \rm It is clear that our theorem generalizes the results in [1-14, 16, 17] since$p>1$. Even for$p=2$, the conditions of Theorem \ref{th1} are different from the conditions in \cite{Antonacci,Bahri,benci} since in \cite{Antonacci} there exist$\beta>2, \alpha>0$such that$H(u)\ge \alpha |u|^{\beta}$for all$u\in \mathbb{R}^N$and in \cite{Bahri,benci} there exists$\mu>2$such that \eqref{1} holds for$t\in [0, T], |x|\ge R$. There are functions satisfying Theorem \ref{th2} but not satisfying \cite{Antonacci, Bahri, benci}. For example, let \begin{gather*} b_i(t)=t, i=1, 2, \dots, N,\quad \mu=3,\\ H(t,u)=\begin{cases} \frac{t}{2T^3}\left(\frac{7}{4}|u|^{\frac{7}{4}}+\sin(|u|-1)+\frac{3}{7}\right),&|u|\ge1, \\[2pt] \frac{|u|^2 t}{2T^3},& |u|<1. \end{cases} \end{gather*} Then$F(t,u)=\frac{B(t)}{3}|u|^{3}+H(t, u)$satisfies Theorem \ref{th2}, but does not satisfies the conditions in \cite{Antonacci,Bahri,benci}. \end{remark} \begin{thebibliography}{00} \bibitem{Antonacci} F.Antonacci, P. 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