\documentclass[twoside]{article} \input amssym.def % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Existence and Multiplicity Results \hfil EJDE--1997/07}% {EJDE--1997/07\hfil Chao-Nien Chen \& Shyuh-yaur Tzeng \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.\ {\bf 1997}(1997), No.\ 07, pp. 1--19. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{\bigskipamount} \\ Existence and Multiplicity Results for Homoclinic Orbits of Hamiltonian Systems \thanks{ {\em 1991 Mathematics Subject Classifications:} 34C37, 49M30, 58E99, 58F09.\newline\indent {\em Key words and phrases:} Hamiltonian system, homoclinic, calculus of variations. \newline\indent \copyright 1997 Southwest Texas State University and University of North Texas.\newline\indent Submitted September 25, 1996. Published March 26, 1997. \newline\indent This work was partially supported by the National Science Council of Republic of China.} } \date{} \author{Chao-Nien Chen \& Shyuh-yaur Tzeng} \maketitle \begin{abstract} Homoclinic orbits play an important role in the study of qualitative behavior of dynamical systems. Such kinds of orbits have been studied since the time of Poincar\'e. In this paper, we discuss how to use variational methods to study the existence of homoclinic orbits of Hamiltonian systems. \end{abstract} \newtheorem{thm}{Theorem} \newtheorem{pro}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{remark}{Remark} \newtheorem{cor}{Corollary} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \section* {Introduction} In the theory of differential equations, a trajectory which is asymptotic to a constant state as $|t|\rightarrow\infty$ is called a homoclinic orbit. Such kinds of orbits have been found in various models of dynamical systems and they frequently have tremendous effect to the dynamics of such nonlinear systems. The homoclinic orbits have been studied since the time of Poincar\'e but mainly by perturbation methods. It is only relatively recently that some new tools have been developed in the calculus of variations to show the existence of homoclinic solutions of nonlinear differential equations. In this article, a class of second order Hamiltonian systems is considered: $$\ddot{q}-L(t)q+V'(t,q)=0 \eqno{(HS)}$$ where $q\in {\Bbb R}^n$. The $n\times n$ matrix $L(t)$ is continuous, symmetric and satisfies the following condition: \begin{description} \item{(L)} There are $\mu_1,\mu_2\in(0,\infty)$ such that $\mu_1|y|^2\leq L(t)y\cdot y\leq \mu_2|y|^2$ for all $t\in{\Bbb R},y\in {\Bbb R}^n$. \end{description} The basic assumption for the function $V(t,y)$ is \begin{description} \item{(V1)} $V\in C^2({\Bbb R}\times{\Bbb R}^n,{\Bbb R})$, $V'(t,0)\equiv D_yV(t,0)=0$, $D^2_yV(t,0)=0$ and $\lim_{|y|\rightarrow 0}\frac{V'(t,y)}{|y|}=0$ uniformly in $t\in{\Bbb R}.$ For any $r>0$, there is a $K=K(r)$ such that $$\sup_{t\in{\Bbb R},\;|y|\leq r}\|D_yV(t,y)\|_\infty+ \|D_y^2V(t,y)\|_\infty\leq K.$$ \end{description} The hypothesis (V1) implies that $0$ is a constant solution of (HS). A non-constant solution of (HS) which tends to 0 as $|t|\rightarrow\infty$ will be called a homoclinic solution or a homoclinic orbit. In \cite{R1}, Rabinowitz proved existence of homoclinic orbits for (HS) when: \begin{description} \item{(P)} $L$ and $V$ are T-periodic in $t$, and \item{(V2)} there is a $\mu>2$ such that $0<\mu V(t,y)\leq y\cdot V'(t,y)$ for all $y\in{\Bbb R}^n\backslash\{0\}$\mbox{ and } $\inf\limits_{|y|=1,\;t\in{\Bbb R}} V(t,y)>0$. \end{description} Subsequently, Coti Zelati and Rabinowitz \cite{CR1} obtained multibump homoclinic solutions for this system. For the first-order-time-periodic Hamiltonian system, the existence of multibump homoclinic solutions was proved by S\'er\'e \cite{S}; there the one-bump solution had been obtained earlier in \cite{CES}. Subsequently there have been further extensions for using variational methods to study such problems in various directions \cite{CM, CR2, GR, M, R3, Sp, St}. The interested reader may consult \cite{R4} for more references. In this paper we intend to investigate the existence of homoclinic orbits of (HS) when L and V are not necessarily periodic in t. The approach to (HS) will involve the use of variational methods of a mini-max nature. Let $E$ be the space $W^{1,2}({\Bbb R},{\Bbb R}^n)$ under the norm $\left( \int_{-\infty}^{\infty}(|\mbox{\it \.q}|^2+|q|^2)dt \right)^{1/2}.$ It is known that $E\subset C^0({\Bbb R},{\Bbb R}^n)$, the space of continuous functions $q$ on ${\Bbb R}$ such that $q(t)\rightarrow 0$ as $|t|\rightarrow\infty.$ The solutions of (HS) are the critical points of the functional $J$ given by \begin{eqnarray*} J(q)&=&\int^{\infty}_{-\infty}\left[\frac{1}{2}(|\mbox{\it\.q}|^2+L(t)q\cdot q)-V(t,q)\right]dt \nonumber \\ &=&\frac{1}{2}\|q\|^2-\int_{-\infty}^{\infty}V(t,q)dt, \end{eqnarray*} where by assumption (L), $$\|q\|^2=\int^{\infty}_{-\infty}(|\mbox{\it\.q}|^2+L(t)q\cdot q)dt$$ is taken as an equivalent norm on $E$. Note that assumption (V1) implies that $J\in C^1(E,{\Bbb R})$. Moreover, critical points of $J$ are classical solutions of (HS), satisfying $\dot{q}(t)\rightarrow 0$ as $|t|\rightarrow\infty$. Thus $q$ is a homoclinic solution of (HS). Let $I^\alpha=\{q\in E|J(q)\leq \alpha\}$. It is not difficult to see from (V1) and (V2) that $V(t,q)=o(|q|^2)$ as $|q|\rightarrow 0$ and $V(t,q)|q|^{-2}\rightarrow\infty$ as $|q|\rightarrow\infty$. Thus $$\label{0.3} J(q)=\frac{1}{2}\|q\|^2+o(\|q\|^2) \quad \mbox{ as } q\rightarrow 0$$ and $I^0\backslash\{ 0 \}$ is non-empty. By the Mountain Pass Theorem $J$ would have a critical value $\beta >0$ given by $$\beta=\inf_{\gamma\in\Gamma}\max_{\theta\in[0,1]}J(\gamma(\theta))\,, \eqno{(MP)}$$ where $$\Gamma=\{\gamma\in C([0,1],E)|\gamma(0)=0 \mbox{ and } \gamma(1)\in I^0\backslash\{0\}\},$$ provided that \begin{description} \item{(PS)} whenever $J(u_m)$ is bounded and $J'(u_m)\rightarrow 0$ as $m\rightarrow\infty,$ the sequence $\{u_m\}$ possesses a (strongly) convergent subsequence in $E$. \end{description} Unfortunately the Palais-Smale condition (PS) is not always satisfied. For example, let $q(t)$ be a homoclinic solution of (HS) obtained in \cite{R1}. Note that by (P) $$J(\tau_mq)=J(q)$$ if $m\in{\Bbb Z}$ and $$\tau_mq(t)=q(t-mT)\,.$$ Therefore, $J(\tau_mq)\rightarrow J(q)>0$ and $J'(\tau_mq)\rightarrow 0$ as $m\rightarrow\infty$. Nevertheless, the sequence $\{\tau_mq\}$ does not possess a convergent subsequence in $E$. The same kind of difficulty occurs in dealing with homoclinic orbits of first order Hamiltonian systems (e.g. \cite{CES,S}), there the authors used the idea of concentration-compactness to treat the Palais-Smale sequences. Although the mini-max structure of (MP) cannot guarantee that there is a critical point $u\in E$ with $J(u)=\beta$, we find a way to justify whether $\beta$ is a critical value of $J$. Our method is based on the following comparison argument. Let $\Omega_k={\Bbb R}\backslash[-k,k],$ and $E_k=W^{1,2}_o(\Omega_k,{\Bbb R}^n)$ with the norm $$\|u\|_k=\left(\int_{\Omega_k}(L(t)u\cdot u+|\dot{u}|^2)dt\right)^{1/2}.$$ Since $w\in E_{k+1}$ can be identified with an element of $E_k$ by extending $w$ to be zero on $\Omega_k\backslash\Omega_{k+1}$, the inclusions $$\label{0.7} E_{k+1}\subset E_k\subset\cdots E$$ will be used without mentioned explicitly, and $J_k$ will be the restriction of $J$ to $E_k$. \paragraph{Definition} A sequence $\{u_{m}\}\subset E$ is called a $\rm (PS)_c$ sequence if $J(u_{m})\rightarrow c$ and $J'(u_{m})\rightarrow 0$ as $m \rightarrow \infty$. Let $\Lambda$ be the set of positive numbers $c$ such that there exists a $\rm (PS)_{c}$ sequence. The set $\Lambda$ in particular contains all the positive critical values of $J$. Let $\delta$ be the infimum of $\Lambda$. It will be shown that $\Lambda$ is a nonempty set and $\delta$ is a positive number. On the restriction $J_{k}$, we define the set $\Lambda_k$ and its infimum $\delta_{k}$ similarly. \begin{thm} There exists a homoclinic solution $q$ of (HS) with $J(q)=\delta$, provided that $\delta\not\in\Lambda_k$ for some $k\in{\Bbb N}$. \end{thm} \begin{remark} We may take $\Omega_k={\Bbb R}\backslash[a_k,b_k],$ where $\{a_k\}$ is a decreasing sequence with $\lim_{k\rightarrow\infty}a_k=-\infty$ and $\{b_k\}$ is an increasing sequence with $\lim_{k\rightarrow\infty}b_k=\infty$. \end{remark} When $\beta>\delta$, $(HS)$ may possess more than one homoclinic orbit. \begin{thm} There exist a homoclinic solution $q$ with $J(q)=\delta$ and a homoclinic solution $\hat q$ with $J(\hat q)=\beta,$ if $\delta<\beta<\delta_k$ for some $k\in{\Bbb N}$. \end{thm} The proofs of Theorems 1 and 2 will be given in section 2. Section 1 contains several preliminary results such as a detailed analysis of Palais-Smale sequences. A sufficient condition for $\delta=\beta$ will be given in section 3. In section 4, some applications of the above theorems will be discussed, including the investigation of perturbations of time periodic Hamiltonian systems. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setcounter{equation}{0} \section{Preliminaries} This section contains several technical results such as various smoothness and qualitative properties of $J$. As mentioned in the introduction, the Mountain Pass Theorem cannot be directly applied to obtain the existence of homoclinic solutions of (HS), since verification of (PS) may not be possible. An alternative approach is to analyze the behavior of Palais-Smale sequences. In doing so, we begin with the Fr\'echet differentiability of $J$. A detailed proof of Proposition 1 can be found in \cite{CR1}. \begin{pro} If V satisfies (V1), then $J\in C^1(E,{\Bbb R})$. \end{pro} Next we prove the boundedness of Palais-Smale sequences. \begin{lemma} If $\{u_m\}$ is a $(PS)_c$ sequence then there is a constant $A$ such that $$\|u_m\|\leq A$$ and $$\|u_m\|_{L^{\infty}}\leq\sqrt{2}\,A\,.$$ \end{lemma} \paragraph{Proof.} Since $J'(u_m)\rightarrow 0$ as $m\rightarrow\infty$, if $m$ is large then $$\|u_m\|^2-\int^{\infty}_{-\infty}V'(t,u_m)\cdot u_mdt=J'(u_m)u_m =o(1)\cdot\|u_m\|\,.$$ Hence \begin{eqnarray} c&=&J(u_m)+o(1) \nonumber \\ &=&J(u_m)-\frac{1}{2}J'(u_m)u_m+o(1)\cdot(1+\|u_m\|) \nonumber \\ &=&\int^{\infty}_{-\infty}[\frac{1}{2}V'(t,u_m)\cdot u_m -V(t,u_m)]dt+o(1)\cdot(1+\|u_m\|) \nonumber \\ &\geq&(\frac{1}{2}-\frac{1}{\mu})\int^{\infty}_{-\infty}V'(t,u_m) \cdot u_mdt+o(1)\cdot(1+\|u_m\|)\,, \end{eqnarray} where the last inequality follows from (V2). Substituting (1.3) into (1.4) yields $$c\geq(\frac{1}{2}-\frac{1}{\mu})\|u_m\|^2+o(1)\cdot(1+\|u_m\|),$$ which completes the proof of (1.1). Then (1.2) follows from the inequality $\|u\|_{L^{\infty}}\leq\sqrt{2}\|u\|$. \begin{cor} If $\{u_m\}$ is a $(PS)_c$ sequence then $$\lim_{m\rightarrow\infty}\sup\|u_m\|\leq \left(\frac{2\mu c}{\mu-2}\right)^{1/2}.$$ \end{cor} The proof of the above corollary follows directly from (1.5) and Lemma 1. \begin{cor} If $q\in E$, and $J'(q)=0$ then $$J(q)\geq\frac{\mu-2}{2\mu}\|q\|^2.$$ \end{cor} \paragraph{Proof.} Note that (1.6) is trivially satisfied when $q\equiv 0$. If $q\not\equiv 0$, (1.6) follows from (1.5) by setting $u_m=q$ for all $m$. \begin{lemma} There exists a $(PS)_\beta$ sequence, where $\beta$ is the number defined in (MP). \end{lemma} This lemma is proved using deformation theory, as Theorem A4 is proven in \cite{R2}. From (MP) and (\ref{0.3}), it is clear that $\beta>0$. Thus Lemma 2 shows that $\Lambda$ is non-empty. \begin{pro} If (V1), (V2) and (L) are satisfied then $\delta>0$. \end{pro} \paragraph{Proof.} Choose $\rho>0$ such that $$|y\cdot V'(t,y)| \leq\frac{\mu_1}{2}|y|^2$$ if $|y|\leq\rho$. Let $\{u_m\}$ be a $(PS)_c$ sequence. Pick $\bar{c}>0$ such that $\left(3\mu\bar{c}/(\mu-2)\right)^{1/2} =\rho/\sqrt{2}$. By Corollary 1 $$\|u_m\|<\left(\frac{3\mu\bar{c}}{\mu-2}\right)^{1/2}$$ if $c\in(0,\bar{c})$ and $m$ is large. Since $\|u_m\|_{L^{\infty}}\leq\sqrt{2}\|u_m\|<\rho$, $$J'(u_m)\frac{u_m}{\|u_m\|}\geq\|u_m\|^{-1}\left(\|u_m\|^2 -\frac{\mu_1}{2}\int^{\infty}_{-\infty} |u_m|^2dt\right)\geq\frac{1}{2}\|u_m\|$$ which implies $\|u_m\|\rightarrow 0$ and consequently $J(u_m)\rightarrow 0$ as $m\rightarrow\infty$. This violates the fact that $\lim_{n\rightarrow\infty}J(u_m)=c>0$. Therefore there is no $(PS)_c$ sequence if $c\in(0,\bar{c})$. So $\delta\geq\bar{c}>0$. Let $\xi:{\Bbb R}\rightarrow[0,1]$ be a $C^{\infty}$-function which satisfies \begin{eqnarray} \xi(t)=\left\{ \begin{array}{l} 0\qquad \mbox{ if } t\in[-(k+1),k+1] \\ 1\qquad \mbox{ if } t\not\in[-(k+2),k+2]\,. \end{array} \right. \end{eqnarray} \begin{lemma} Let $\{u_m\}$ be a $(PS)_c$ sequence. Assume there is an increasing sequence $\{t_m\}$ such that $t_m\rightarrow\infty$ and $$\int^{t_m}_{-t_m}|u_m|^2dt\rightarrow 0$$ as $m\rightarrow\infty$. Let $w_m$ be the restriction of $\xi u_m$ on $\Omega_k$. Then $w_m\in E_k$ and $J_{k}(w_m)\rightarrow c$ and $J'_k(w_m)\rightarrow 0$ as $m\rightarrow\infty.$ \end{lemma} \paragraph{Proof.} It suffices to show that $$\lim_{m\rightarrow\infty}|J_k(w_m)-J(u_m)|=0$$ and $$\lim_{m\rightarrow\infty}\sup_{\|\phi\|_k\leq 1}|J'_k(w_m)\phi- J'(u_m)\phi|=0\,.$$ By a direct computation, \begin{eqnarray} J'_k(w_m)\phi-J'(u_m)\phi&=&\int_{t\geq|k|}[L(t)(\xi(t)-1)u_m\cdot\phi +(\xi(t)-1)\dot{u}_m\dot{\phi} \nonumber \\ &&+u_m\dot{\xi}\dot{\phi}+(V'(t,w_m)-V'(t,u_m))\cdot\phi]dt \end{eqnarray} Let $Q=[-(k+2),-k]\cup[k,k+2].$ Applying Schwarz inequality yields \begin{eqnarray} \left|\int_{t\geq|k|}L(t)(\xi(t)-1)u_m\cdot\phi dt\right| &\leq& \left(\mu_2\int_Q|u_m|^2dx\right)^{1/2}, \\ \left|\int_{t\geq|k|}u_m\dot{\xi}\dot{\phi}dt\right| &\leq&\|\dot{\xi}\|_{L^{\infty}}\left(\int_Q|u_m|^2dx\right)^{1/2},\\ \left|\int_{t\geq|k|}(\xi(t)-1)\dot{u}_m\dot{\phi}dt\right| &\leq&\left(\int_Q|\dot{u}_m|^2dt\right)^{1/2}, \end{eqnarray} and \begin{eqnarray} \lefteqn{\left|\int_{t\geq|k|}(V'(t,w_m)-V'(t,u_m))\cdot\phi dt\right|} &&\nonumber\\ &\leq& 2A_1\int_Q|u_m|| \phi|dt \\ &\leq&2A_1\left(\int_Q|\phi|^2dt\right)^{1/2} \left(\int_Q|u_m|^2dt\right)^{1/2},\nonumber \end{eqnarray} where $A_1=\max\{\left|D^2_yV(t,y)\right|:\, |y|\leq \sqrt2\,A,\; t\in Q\}$, by making use of Lemma 1. Since $Q\subset[-t_m,t_m]$ if $m$ is large, (1.10) follows from (1.11)-(1.15) and (1.8), provided that $$\int_Q|\dot u_m|^2dt\rightarrow 0\quad\mbox{ as }m\rightarrow\infty.$$ We now prove (1.16). Note that by passing to a subsequence if necessary we may assume $t_{m+1}-t_m>2$. Let $\xi_m$ be a $C^\infty_0({\Bbb R})$ function which satisfies $0\leq\xi_m\leq 1$, $|\dot\xi_m|\leq 1$ and $$\xi_m(t)=\left\{ \begin{array}{l} 1 \qquad \mbox{ if } t\in[-t_m,t_m] \\ 0 \qquad \mbox{ if } t\not\in(-t_{m+1},t_{m+1})\,. \end{array} \right.$$ Then by Lemma 1 there is a $C_1>0$ such that $$\|\xi_mu_m\|^2\leq\int^\infty_{-\infty}\mu_2|u_m|^2dt+2\int^\infty_{-\infty}|u_m|^2|\dot\xi_m|^2dt +2\int_{-\infty}^\infty\xi_m^2|\dot u_m|^2dt\leq C_1\,.$$ If $m$ is large then \begin{eqnarray} &&\int^{t_m}_{-t_m}L(t)u_m\cdot\xi_mu_mdt+\int^{t_m}_{-t_m}\dot\xi_mu_m\cdot\dot u_mdt -\int^{t_m}_{-t_m}V'(t,u_m)\xi_mu_mdt \nonumber\\ &&+\int^{t_m}_{-t_m}\xi_m|\dot u_m|^2dt=J'(u_m)\xi_mu_m=o(1). \end{eqnarray} Arguing like above and using (1.8), we conclude that the first three integrals of (1.17) tend to zero as $m\rightarrow\infty$ and consequently $$\int_Q|\dot u_m|^2dt\leq\int^{t_{m-1}}_{-t_{m-1}}|\dot u_m|^2dt\leq \int^{t_m}_{-t_m}\xi_m|\dot u_m|^2dt\rightarrow 0\mbox{ as }m\rightarrow\infty.$$ Observe that \begin{eqnarray*} J_k(w_m)-J(u_m)&=&\int_Q\frac{1}{2}[(\xi^2-1)(L(t)u_m\cdot u_m+|\dot{u}_m|^2) \\ &&+|\dot{\xi}|^2|u_m|^2+2\xi\dot{\xi}u_m\cdot\dot{u}_m]+[V(t,u_m)-V(t,w_m)]dt \\ &&-\int^k_{-k}[\frac 12(L(t)u_m\cdot u_m+|\dot u_m|^2)-V(t,u_m)]dt. \end{eqnarray*} Thus (1.9) follows from several estimates which are similar to the above. The next two lemmas indicate the relationship between Palais-Smale sequences and critical points of $J$. We refer to \cite{CR1} for detailed proofs. \begin{lemma} Let $\{u_m\}$ be a $(PS)_c$ sequence. Then there exists a subsequence $\{u_{m_j}\}$ such that $$u_{m_j}\rightarrow u \mbox{ weakly in } E \mbox{ and strongly in } L^{\infty}_{loc}({\Bbb R}, {\Bbb R}^n)$$ for some $u\in E$ which satisfies $J'(u)=0$ and $J(u)\leq c$. \end{lemma} \begin{lemma} Let $\{u_m\}$ be a $(PS)_c$ sequence. Assume $\{u_m\}$ converges to $u\in E$ both weakly in $E$ and strongly in $L^{\infty}_{loc}({\Bbb R},{\Bbb R}^n).$ If $v_m=u_m-u$, then $$\lim_{m\rightarrow\infty}J'(v_m)=0 \quad \mbox{ and } \lim_{m\rightarrow\infty}J(v_m)=c-J(u)\,.$$ \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setcounter{equation}{0} \section{Existence results} We now prove the existence of homoclinic orbits of $(HS)$. \begin{thm} Suppose there exists a $(PS)_c$ sequence such that $c>0$ and \begin{eqnarray} c\not\in\Lambda_k \end{eqnarray} for some $k\in{\Bbb N}$, then there is a homoclinic solution $q$ of (HS) and \begin{eqnarray} c\geq J(q)\geq\delta. \end{eqnarray} \end{thm} \paragraph{Proof} Let $\{u_m\}$ be a $(PS)_c$ sequence. By Lemma 4, there exist a $q\in E$ and a subsequence, still denoted by $\{u_m\}$, such that \begin{eqnarray} &u_m\rightarrow q \quad \mbox{ weakly in } E \mbox{ and strongly in } L^{\infty}_{loc}({\Bbb R},{\Bbb R}^n),&\nonumber \\ & J'(q)=0, \mbox{ and}& \\ &J(q)\leq c\,.& \end{eqnarray} We claim that $q\not\equiv 0$. This is true if there exist $k,l\in {\Bbb N}$ and $b>0$ such that if $m\geq l$ then $$\int_{-k}^k|u_m|^2dt\geq b.$$ Suppose (2.5) is false. Then there exist a sequence $\{t_m\}$ with $\lim_{m\rightarrow\infty}t_m=\infty$, and a subsequence, still denoted by $\{u_m\}$, such that $$\lim_{m\rightarrow\infty}\int_{-t_m}^{t_m}|u_m|^2dt=0\,.$$ Let $\xi$ be defined as in (1.7) and $w_m$ be the restriction of $\xi u_m$ to $\Omega_k.$ Invoking Lemma 3 yields $$c\in\Lambda_k$$ which contradicts (2.1). Therefore (2.5) holds and $q\not\equiv 0$. This together with (2.3) and Corollary 2 shows that $J(q)>0$. Moreover (2.2) follows from the definition of $\delta$ and (2.4). Since $E\subset C^0({\Bbb R},{\Bbb R}^n),q(t)\rightarrow 0$ as $|t|\rightarrow\infty$. This together with (HS) and (V1) shows that $\ddot{q}\in L^2({\Bbb R},{\Bbb R}^n)$. Then $q\in W^{2,2}({\Bbb R},{\Bbb R}^n)$ implies $\dot{q}\in C^0({\Bbb R},{\Bbb R}^n)$. Thus $q$ is a homoclinic solution of $(HS)$.\\ \paragraph{Proof of Theorem 1.} By the definition of $\Lambda$ and Proposition 2, there is a $(PS)_\delta$ sequence. Applying Theorem 3 gives a homoclinic solution $q$ of $(HS)$ with $J(q)=\delta$. \paragraph{Proof of Theorem 2.} Since $\delta\not\in\Lambda_k,$ by Theorem 1 there is a homoclinic solution $q$ of (HS) with $J(q)=\delta.$ Invoking Lemma 2 and Lemma 3, we get a $(PS)_\beta$ sequence $\{u_m\}$ which converges to $\hat q$ both weakly in $E$ and strongly in $L^{\infty}_{loc}({\Bbb R},{\Bbb R}^n)$. Moreover $J'(\hat q)=0$ and $J(\hat q)\leq\beta$. Since $\beta\not\in\Lambda_k$, it follows from the same reasoning as the proof of Theorem 3 that $\hat q$ is a homoclinic solution of (HS). Suppose $\hat q=q$. Setting $v_m=u_m-q$, we see from Lemma 5 that $\{v_m\}$ is a $(PS)_{\beta-\delta}$ sequence. Since $0<\beta-\delta<\delta_k$, repeating the above arguments shows that $\{v_m\}$ converges weakly to some $\bar{v}\in E\backslash\{0\}$. This contradicts the fact that $u_m$ converges weakly to $q$. So $\hat q\neq q$. \begin{remark} This proof shows that Theorem 2 still holds if $\delta<\beta$ and $\beta\not\in\Lambda_k$, $\delta\not\in\Lambda_j$, $\beta-\delta\not\in\Lambda_i$ for some $i,j,k\in{\Bbb N}$. \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setcounter{equation}{0} \section{A sufficient condition for $\delta=\beta$} Although it has been shown that $\delta>0,$ in general it seems to be difficult to obtain an optimal lower bound for $\delta.$ Here we illustrate a condition which ensures $\delta=\beta:$ \begin{description} \item{(V3)} For all $t\in{\Bbb R},|y|=1$, $\rho^{-1}V'(t,\rho y)\cdot y$ is an increasing function of $\rho$ if $\rho\in(0,\infty)$. \end{description} \begin{pro} If (V1)-(V3) and (L) are satisfied then $\delta=\beta$. \end{pro} To Prove Proposition 3, we need the following proposition. \begin{pro} If (V1)-(V3) and (L) are satisfied then $$\beta=\inf_{u\in E \atop u\not\equiv 0}\max_{\theta\in[0,\infty)} J(\theta u).$$ \end{pro} Since the proof of this proposition is similar to that of Proposition 2.14 in \cite{DN}, we omit it. \begin{cor} If (V1)-(V3) and (L) are satisfied, then $\beta\leq\beta_k\leq\beta_{k+1},$ where $$\beta_k=\inf_{u\in E_k\atop u\not\equiv 0} \max_{\theta\in[0,\infty)}J_k(\theta u)\,.$$ \end{cor} The proof of this corollary follows easily from (\ref{0.7}). \paragraph{Proof of Proposition 3.} It suffices to show that $\delta\geq\beta$, since the reversed inequality is always true. Let $\{u_m\}$ be a $(PS)_c$ sequence with $c>0$. Then there is an $\epsilon_1>0$ such that for large $m$ $$\|u_m\|\geq\epsilon_1.$$ For $u_m\not\equiv 0,$ we set $$g_m(\rho)=J(\rho u_m).$$ It is clear that $g_m(0)=0.$ Since $$g'_m(\rho)=\rho\|u_m\|^2-\int_{-\infty}^{\infty}V'(t,\rho u_m)\cdot u_mdt,$$ it follows from (V1) that $g'_m(\rho)>0$ if $\rho$ is a sufficiently small positive number. Moreover, we know from (V2) that $$\lim_{\rho\rightarrow\infty}g_m(\rho)=-\infty.$$ Hence there is a $\rho_m\in(0,\infty)$ such that $$g'_m(\rho_m)=0$$ and $$g_m(\rho_m)=\max_{\rho\in[0,\infty)}g_m(\rho)\,.$$ By Proposition 4 $$\beta\leq g_m(\rho_m)\,.$$ Let $R(z)=\{t\in{\Bbb R}||t-z|\leq\frac{1}{2}\}$. We claim there exist a sequence $\{z_m\}\subset{\Bbb Z}$ and an $\epsilon_2>0$ such that $$\|u_m\|_{L^{\infty}(R(z_m))}\geq\epsilon_2\,.$$ Suppose (3.8) is false. Then there is a subsequence, still denoted by $\{u_m\},$ such that $$\sup_{z\in{\Bbb Z}}\|u_m\|_{L^{\infty}(R(z))}\rightarrow 0 \quad \mbox{ as } m\rightarrow\infty\,.$$ Given $\epsilon_3>0$, by (V1) there is an $s>0$ such that $$|V'(t,y)\cdot y|\leq\epsilon_3|y|^2\quad\mbox{ for }|y|\leq s\,.$$ If $m$ is large, (3.9) and (3.10) imply that $$\left|\int_{-\infty}^{\infty}V'(t,u_m)\cdot u_mdt\right|\leq\epsilon_3\int_{-\infty} ^{\infty}|u_m|^2dt.$$ Since $$\|u_m\|^2=J'(u_m)u_m+\int^{\infty}_{-\infty}V'(t,u_m)\cdot u_mdt\leq o(1)\|u_m\|+\epsilon_3\|u_m\|^2,$$ it follows from Lemma 1 that $$\|u_m\|^2\leq\frac{1}{4}\epsilon_1^2+A^2\epsilon_3\,.$$ Choosing $\epsilon_3<\epsilon_1^2/(4A^2)$ yields $$\|u_m\|^2<\frac{1}{2}\epsilon_1^2$$ which contradicts (3.2). Therefore (3.8) must hold. Let $v_m(t)=u_m(t+z_m)$. Since $\|v_m\|$ is bounded, there is a subsequence, still denoted by $\{v_m\}$, such that $$v_m\rightarrow\bar{v}\mbox{ in } L^{\infty}_{loc}({\Bbb R},{\Bbb R}^n)$$ and $$\|\bar{v}\|_{L^{\infty}[-1/2, 1/2]}\geq\epsilon_2\,.$$ Since $v_m$ and $\bar{v}$ are continuous on $[-1/2,1/2]$, there is a subinterval $[a,b]$ of $[-1/2,1/2]$ such that $$\min_{t\in[a,b]}|v_m(t)|\geq\epsilon_4 \mbox{ for all large m,}$$ where $\epsilon_4=\epsilon_2/2$. By (3.4) and (3.5) $$\|u_m\|^2=\frac{1}{\rho_m}\int^{\infty}_{-\infty}V'(t,\rho_mu_m)\cdot u_mdt.$$ Let $D_m=[a+z_m,b+z_m]$. Since (V2) implies that $$\frac{y\cdot V'(t,y)}{|y|^2}\geq\frac{\mu V(t,y)}{|y|^2}\rightarrow \infty\quad \mbox{ uniformly in t as } |y|\rightarrow\infty,$$ We get \begin{eqnarray} \|u_m\|^2&\geq&\frac{1}{\rho_m}\int_{D_m}V'(t,\rho_mu_m(t)) \cdot u_m(t)dt \nonumber\\ &=&\int_{D_m}\rho_m^{-2}|u_m(t)|^{-2}V'\left(t,\rho_m|u_m(t)| \frac{u_m(t)}{|u_m(t)|}\right)\times \nonumber\\ & &\rho_m|u_m(t)|\frac{u_m(t)}{|u_m(t)|}|u_m(t)|^2dt\nonumber\\ &\geq&\epsilon^2_4\int_{D_m}\rho^{-2}_m|u_m(t)|^{-2}V' \left(t,\rho_m|u_m(t)|\frac{u_m(t)}{|u_m(t)|}\right)\times \nonumber\\ & &\rho_m|u_m(t)|\frac{u_m(t)}{|u_m(t)|}dt\rightarrow\infty\mbox{ as } \rho_m\rightarrow\infty. \end{eqnarray} It follows from Lemma 1 that $\{\rho_m\}$ is bounded. Let $$h(\rho)=\frac 12\rho^2\int^{\infty}_{-\infty}V'(t,u_m)\cdot u_mdt-\int^{\infty}_{-\infty} V(t,\rho u_m)dt.$$ Since $$h'(\rho)=\rho\int^{\infty}_{-\infty}V'(t,u_m)\cdot u_mdt- \int^{\infty}_{-\infty}V' (t,\rho u_m)\cdot u_mdt,$$ it follows from (V3) that $h'(\rho)>0$ if $\rho\in(0,1)$ and $h'(\rho)<0$ if $\rho\in(1,\infty).$ Therefore, $$h(1)=\max_{\rho\in[0,\infty)}h(\rho).$$ Since \begin{eqnarray*} g_m(\rho_m)&=&\frac 12\rho^2_mJ'(u_m)u_m+\frac 12\rho_m^2\int^{\infty}_{-\infty}V'(t,u_m) \cdot u_mdt \\ &&-\int^{\infty}_{-\infty}V(t,\rho_mu_m)dt=h(\rho_m)+o(1), \end{eqnarray*} we have \begin{eqnarray*} \beta&\leq&\lim_{m\rightarrow\infty}\inf g_m(\rho_m) \\ &\leq&\lim_{m\rightarrow\infty}\inf\int_{-\infty}^{\infty}[\frac 12V'(t,u_m)\cdot u_m -V(t,u_m)]dt \\ &=&\lim_{m\rightarrow\infty}\inf[\frac 12\|u_m\|^2-\int^{\infty}_{-\infty}V(t,u_m)dt -\frac 12J'(u_m)u_m] \\ &=&\lim_{m\rightarrow\infty}J(u_m)=c. \end{eqnarray*} Since $c$ is arbitrary, it follows that $\beta\leq\delta.$ This completes the proof. \begin{remark} Under the hypothesis of Proposition 3, it follows that $$\delta_k=\beta_k\,.$$ and $$\delta_k\leq\delta_{k+1}\,.$$ \end{remark} In this case, Theorem 1 can be recast as follows. \begin{thm} There exists a homoclinic solution of (HS) if $$\beta<\lim_{k\rightarrow\infty}\delta_{k}.$$ \end{thm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setcounter{equation}{0} \setcounter{thm}{4} \section{Examples} In this section some existence and nonexistence results for the homoclinic orbits of (HS) will be discussed. For the time-periodic Hamiltonian system of the form (HS), it has been shown (e.g. \cite{R1,CR1}) that there exists at least one homoclinic orbit. \begin{pro} If (L) (P) (V1) and (V2) are satisfied, then (HS) possesses a homoclinic orbit $q\in W^{1,2}({\Bbb R},{\Bbb R}^n)$ such that $J(q)=\delta$. \end{pro} \begin{remark} If (V3) is also satisfied, then by Proposition 3 there is a homoclinic orbit $q$ of (HS) with $J(q)=\beta.$ \end{remark} \begin{lemma} If (L),(P) and (V1)-(V3) are satisfied then $\beta_k=\beta$ for all $k$. \end{lemma} \paragraph{Proof.} As noted in Remark 1, we may take $\Omega_k={\Bbb R}\backslash [-kT,kT]$. Let $q$ be a homoclinic orbit of (HS) with $J(q)=\beta$. If $$u_m(t)= \left\{ \begin{array}{ll} q(t-mT) & \mbox{ for } t\in[(k+1)T,\infty) \\ \frac{t-kT}{T}q((k-m+1)T) & \mbox{ for } t\in[kT,(k+1)T) \\ 0 & \mbox{ for } t\in (-\infty,-kT], \end{array} \right.$$ then by a direct computation $$\lim_{m\rightarrow\infty}\max_{\theta\in[0,\infty)} J_k(\theta u_m)=\beta.$$ Since $\lim_{\theta\rightarrow\infty}J_k(\theta u_m)=-\infty,$ (4.1) implies $$\beta_k\leq\beta.$$ On the other hand, $W^{1,2}_0(\Omega_k,{{\Bbb R}^n})\subset W^{1,2}({\Bbb R},{\Bbb R}^n)$ yields $$\beta_k\geq \beta.$$ Combining (4.2) with (4.3) gives $\beta_k=\beta.$ In the use of comparison arguments in what follows, we let $$\ddot q-\widetilde L(t)q+\widetilde V'(t,q)=0 \eqno{(HS)^\sim}$$ be a Hamiltonian system having the same form as (HS). The corresponding functional associated with (HS)$^\sim$ is $$\widetilde J(q)=\int^\infty_{-\infty} \left[ \frac 12 \left( |\dot q|^2+\widetilde L(t)q\cdot q \right) -\widetilde V\left(t,q\right) \right ]dt.$$ Here $\widetilde V$ and $\widetilde L$ as well as $V$ and $L$ are assumed to satisfy (V1), (V2) and (L). In the same way as $\beta$ is defined in (MP), let $\widetilde\beta$ be the mountain pass minimax value of $\widetilde J$. Similarly, we define $\widetilde\delta$, $\widetilde J_k$, $\widetilde\beta_k$, and $\widetilde\delta_k$ by the same manner. It is clear from (MP) that $$\widetilde\beta\leq\beta$$ if $$(L(t)-\widetilde L(t))y\cdot y\geq 0$$ and $$\widetilde V(t,y)\geq V(t,y)$$ for all $t\in{\Bbb R}$ and $y\in{\Bbb R}^n$. We now investigate the existence of homoclinic orbits of (HS) when a time-periodic Hamiltonian system is perturbed. Let $A=\{y|\alpha<|y|<\widehat\alpha\}$, where $0\leq\alpha<\widehat\alpha$. Set $B=(a,b)\times A$, where $-\infty&V(t,y)\quad\mbox{ for }(t,y)\in B \\ \widetilde V(t,y)&=&V(t,y)\quad \mbox{ for }(t,y)\in {{\Bbb R}^{n+1}}\backslash B \end{eqnarray} \paragraph{Example 1.} The perturbed system (HS)$^\sim$possesses at least one homoclinic solution$\widetilde q$. Indeed, by Proposition 5 and Remark 4 there is a homoclinic solution$q_1$of (HS) with$J(q_1)=\beta.$If the set${\cal L}=\{(t,q_1(t))|t\in{\Bbb R}\}$does not intersect$B$then$q_1$is also a homoclinic solution of (HS)$^\sim$. Suppose${\cal L}\cap B\neq\emptyset$. Then by (4.7) and (4.8), $$\widetilde J(q_1)\beta which contradicts (4.13). \paragraph{Example 4.} When \widetilde V satisfies (4.9) and (4.10), the perturbed system (HS)^\sim may possess a homoclinic solution \widetilde q with \widetilde J(\widetilde q)>\widetilde\beta. Here we construct an example as follows. Let W(t,y) be a function which satisfies $$W(t,y) < V(t,y)\quad\mbox{ for } (t,y)\in B\,,$$ and $$W(t,y) = V(t,y)\quad\mbox{ for } (t,y)\in([0,T]\times {{\Bbb R}^n})\backslash B\,,$$ where B=(a,b)\times A, A=\{y|\alpha<|y|<\infty\}, 0<\alpha and 00 and \lim_{|t|\rightarrow\infty}|\widetilde q(t)|=0, the set {{\cal L}_2}=\{(t,\widetilde q(t))|t\in{\Bbb R}\} intersects only a finite number of B_l. Let \widetilde B be the union of B_l for which B_l\cap {\cal L}_2\neq\emptyset. Define \widetilde V(t,y) by$$ \widetilde V(t,y)= \left\{ \begin{array}{ll} W(t,y) & \mbox{ if } (t,y)\in\widetilde B. \\ V(t,y) & \mbox{ if } (t,y)\in{\Bbb R}^{n+1}\backslash\widetilde B. \end{array} \right. $$Clearly, \widetilde q is a homoclinic solution of (HS)^\sim. Moreover, q_1 is also a homoclinic solution of (HS)^\sim if q_1(t)=q(t+jT) and j is sufficiently large. Consequently \widetilde\beta=\beta. Finally the fact that {\cal L}_2 intersects B_l shows that \widetilde J(\widetilde q)>\beta=\widetilde\beta. \begin{remark} In Examples 1-4, all these perturbations can be made arbitrarily \lq\lq small". \end{remark} \paragraph{Example 5.} If V satisfies (V3), and there exist t_0\in{\Bbb R} and T>0 such that for all y\in{\Bbb R}^n \begin{eqnarray} V(t,y)\geq V(t+T,y),&& (L(t)-L(t+T))y\cdot y\leq 0 \quad\mbox{if }t\geq t_0 \\ V(t,y)\geq V(t-T,y),&& (L(t)-L(t-T))y\cdot y\leq 0 \quad\mbox{if }t< t_0+T, \end{eqnarray} then (HS) possesses a homoclinic solution q with J(q)=\beta. To show this, we let \widetilde V(t,y)=V(t,y), \widetilde L(t)=L(t) for t\in[t_0,t_0+T], y\in{\Bbb R}. Let \widetilde V and \widetilde L satisfy (P). By Proposition 5 and Remark 4, (HS)^\sim possesses a homoclinic solution \widetilde q with \widetilde J(\widetilde q)=\widetilde\beta. Let$${\cal L}_3=\{(t,y)|\widetilde V(t,y)>V(t,y)\mbox{ or }(\widetilde L(t)-L(t))y\cdot y<0)\}\,.$$If the set {\cal L}_4=\{(t,\widetilde q(t))|t\in{\Bbb R}\} does not intersect {\cal L}_3 then \widetilde q is also a homoclinic solution of (HS). Otherwise, we may proceed as in Example 1 to get \beta<\widetilde\beta. Then \delta_k>\beta for some k and consequently (HS) possesses a homoclinic solution q with J(q)=\beta, provided that $$\delta_k\geq\widetilde\beta.$$ Indeed, by Lemma 6, $$\widetilde\beta_k=\widetilde\beta.$$ for all k. Moreover it follows from (3.14) that $$\delta_k=\beta_k.$$ Since (4.16) and (4.17) imply that $$\beta_k\geq\widetilde\beta_k,$$ putting (4.19)-(4.21) together gives (4.18). \begin{remark} The results of Example 5 still hold if instead of (V3), V satisfies \begin{description} \item{(V3)'} There is a t_1>0 such that, for all t\in(-\infty,-t_1]\cup[0,T]\cup[t_1,\infty) and |y|=1, \rho^{-1}V'(t,\rho y)\cdot y is an increasing function of \rho if \rho\in(0,\infty). \end{description} \end{remark} \paragraph{Example 6.} Consider (HS) where L satisfies (P) and there is a t_0\in{\Bbb R} such that for all y\in{\Bbb R}^n \begin{eqnarray} V(t,y)&=&V(t-T,y)\quad\mbox{ if }t\leq t_0\nonumber \\ V(t,y)&=&V(t+T,y)\quad\mbox{ if }t\geq t_0\,. \end{eqnarray} Set \widetilde L=L. Without loss of generality, we may assume that t_0=0. Let \beta_+=\widetilde\beta, when $$\widetilde V\mbox{ satisfies (P) and }\widetilde V(t,y)=V(t,y) \mbox{ for }t\geq 0.$$ Let \beta_{\_}=\widetilde\beta, when $$\widetilde V\mbox{ satisfies (P) and }\widetilde V(t,y)=V(t,y) \mbox{ for }t\leq 0$$ If \beta_{\_}\leq\beta_+, V satisfies (V3) and there is an \alpha_1>0 such that $$V(t,y)>V(t-T,y)\qquad\mbox{ for }t\in(0,T)\mbox{ and }|y|<\alpha_1,$$ then (HS) possesses a homoclinic solution q. This can be done by proving \beta<\beta_{\_}, since \delta_k=\beta_{\_} if k is large. When (4.24) holds, let \widetilde q be a homoclinic solution of (HS)^\sim with \widetilde J(\widetilde q)=\beta_{\_}. Let \theta_0>0 be fixed such that$$\widetilde J(\theta_0\widetilde q)<0.$$If t is sufficiently large then \theta_0|\widetilde q(t)|<\alpha_1. Let q_1(t)=\widetilde q(t+lT) and choose l large enough such that \theta_0|q_1(t)|=\theta_0|\widetilde q(t+lT)|<\alpha_1 for t\geq 0. Then invoking (4.25) yields$$J(\theta q_1)<\widetilde J(\theta q_1)\quad\mbox{ for }\theta \in[0,\theta_0]\,.$$Hence$$\max_{\theta\in[0,\theta_0]}J(\theta q_1)<\max_{\theta \in[0,\theta_0]}\widetilde J(\theta q_1)=\beta_{\_}$$and$$J(\theta_0 q_1)<\widetilde J(\theta_0 q_1)<0\,.$$It follows from (MP) that \beta<\beta_{\_}. \begin{remark} \begin{description} \item{(a)} It is easy to see that in Example 6 (HS) still possesses a homoclinic solution if (4.25) is replaced by$$V(t,y)\geq V(t-T,y)\mbox{ for }t\in[0,T]\mbox{ and }|y|<\alpha_1. $$\item{(b)} As a similar case to Example 6, we may consider \begin{eqnarray*} L(t)=L(t-T),&& V(t,y)=V(t-T,y)\mbox{ if }t\leq t_0\\ L(t)=L(t+T_1),&& V(t,y)=V(t+T_1,y)\mbox{ if }t\geq t_0, \end{eqnarray*} where T_1>0. \end{description} \end{remark} As illustrated in the above examples, Proposition 3 has been applied as a good way to obtain an optimal lower bound for \delta_k if condition (V3) is satisfied. In the next example, our aim is to find estimates of \delta_k to fulfill the requirement of Theorem 1 in the situation where (V3) does not hold. \paragraph{Example 7.} Let V(t,y)=H(t)\hat V(y), where \inf_{t\in{\Bbb R}}H(t)>0. Let \beta_0 be the mountain pass minimax value of J on the subspace W^{1,2}_0((-k,k),{\Bbb R}^n) of E. Clearly$$ \beta_0\geq\beta\,.$$In view of the proof of Proposition 2, \bar c=(\mu-2)\rho^2/(6\mu) is a lower bound of \delta_k, where \rho can be chosen as large as possible; so that, for |t|\geq k, if |y|\leq\rho then$$ |H(t)y\cdot\hat V'(y)|\leq\frac{\mu_1}{2}|y|^2\,.$$Since this inequality holds when \sup_{|t|\geq k}H(t)\leq\lambda and$$ \frac{|y\cdot\hat V'(y)|}{|y|^2}\leq\frac{\mu_1}{2\lambda}\,,$$we see that$\bar c>\beta_0$and consequently$\delta_k>\beta$if$\lambda\$ is small enough. \begin{thebibliography}{12} \bibitem{AR} A. Ambrosetti and P.H. Rabinowitz, Dual Variational methods in critical point theory and applications, J. Funct. 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Strobel, Multibump solutions for a class of periodic Hamiltonian systems, University of Wisconsin, Thesis, 1994. \end{thebibliography} \bigskip \noindent{\sc Chao-Nien Chen \& Shyuh-yaur Tzeng \\ Department of Mathematics \\ National Changhua University of Education \\ Changhua, Taiwan, ROC } \\ E-mail address: chenc@math.ncue.edu.tw \end{document}