\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Infinitely many homoclinic orbits \hfil EJDE--1999/42} {EJDE--1999/42\hfil Cheng Lee \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 1999}(1999), No.~42, pp. 1--12. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Infinitely many homoclinic orbits for Hamiltonian systems with group symmetries \thanks{ {\em 1991 Mathematics Subject Classifications:} 34B30, 34C37. \hfil\break\indent {\em Key words and phrases:} Hamiltonian system, homoclinic orbits. \hfil\break\indent \copyright 1999 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted March 19, 1999. Published October 12, 1999. \hfil\break\indent Sponsored by the National Science Council of Taiwan (NSC 89-2115-M-018-011)} } \date{} % \author{Cheng Lee} \maketitle \begin{abstract} This paper deals via variational methods with the existence of infinitely many homoclinic orbits for a class of the first-order time-dependent Hamiltonian systems $$ \dot{z}=JH_z(t,z) $$ without any periodicity assumption on $H$, providing that $H(t,z)$ is G-symmetric with respect to $z\in {\mathbb R}^{2N}$, is superquadratic as $|z|\to\infty$, and satisfies some additional assumptions. \end{abstract} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \section{Introduction} This paper is an extension of the work [7]. We consider the existence of infinitely many homoclinic orbits for the first-order time-dependent Hamiltonian systems $$ \dot{z}=JH_z(t,z), \eqno(HS) $$ where $z=(p,q)\in{\mathbb R}^{2N}$, $H\in C^1({\mathbb R}\times {\mathbb R}^{2N},{\mathbb R})$, $H(t,0)\equiv 0$, and $J$ is the standard symplectic structure on ${\mathbb R}^{2N}$, \[ J=\left(\begin{array}{lll} 0 & -I_N\\ I_N & 0 \end{array}\right) \] with $I_N$ being the $N\times N$ identity matrix. By a homoclinic orbit we mean a solution $z\in C^1({\mathbb R},{\mathbb R}^{2N})$ of (HS) which satisfies $z(t)\not\equiv 0$ and the asymptotic condition $z(t)\to 0$ as $|t|\to\infty$. Establishing the existence of homoclinic orbits for systems like (HS) is a classical problem. Up to 1990, apart from a few isolated results, the only method for dealing with such a problem was the small-perturbation technique of Melnikov. In very recent years this kind of problem has been deeply investigated through variational methods pioneered by Rabinowitz, Coti-Zelati, Ekeland, S$\acute{\rm e}$r$\acute{e}$, Hofer, Wysocki and others, see [2,4-6,8,11-12,14-18]. These papers considered Hamiltonians for the first-order systems (HS) of the form \[ H(t,z)=\frac{1}{2}Az\cdot z+R(t,z), \] where $A$ is a $2N\times 2N$ symmetric and constant matrix such that each eigenvalue of JA has a nonzero real part, and $R(t,z)$ is periodic in $t$ and globally superquadratic in $z$. They showed that (HS) has at least one homoclinic orbit. The existence of infinitely many homoclinic orbits of (HS) was also established in [16,17] if, in addition, $R(t,z)$ is convex in $z$. Recall that, for the particular case of second order systems of the type \[ -\ddot{q}=-L(t)q+W_q(t,q), \] where $L\in C({\mathbb R},{\mathbb R}^{N^2})$ is a symmetric matrix-valued function, the works [15] (among other results) and [6,12] obtained some existence results for homoclinic orbits without periodicity assumptions on the Hamiltonian \[ H(t,p,q)=\frac{1}{2}|p|^2-\frac{1}{2}L(t)q\cdot q+W(t,q)\quad (p,q)\in{\mathbb R}^{2N}, \] providing instead that the smallest eigenvalue of $L(t)$ grows without bound as $|t|\to\infty$, and $W(t,q)$ satisfies some growth assumptions. Motivated by the works of [6,12,15]\,\, Ding and Li studied in [9] the Hamiltonian \renewcommand{\theequation}{1.\arabic{equation}} \setcounter{equation}{0} \begin{equation} H(t,z)=-\frac{1}{2}M(t)z\cdot z+R(t,z), \end{equation} where \[ M(t)=\left(\begin{array}{lll} 0 & L(t)\\ L(t) & 0 \end{array}\right), \] with $L$ being an $N\times N$ symmetric matrix-valued function. They proved that (HS) has at least one homoclinic orbit under the assumptions: (L$_1$) The smallest eigenvalue of $L(t)$ approaches $\infty$ as $|t|\to\infty$, i.e., \hspace{1.55cm} $l(t)\equiv \inf_{\xi\in{\mathbb R}^N,|\xi|=1} L(t)\xi\cdot \xi\to\infty$ as $|t|\to\infty$; (L$_2$) $L\in C({\mathbb R},{\mathbb R}^{N^2})$ and there exists $T_0>0$ such that $2L(t)\underline{+}\frac{d}{dt}L(t)$ are nonnegative definite for all $|t|\geq T_0$; (R$_1$) $R\in C^1({\mathbb R}\times{\mathbb R}^{2N},{\mathbb R})$ and there exists $\mu>2$ such that \[ 0<\mu R(t,z)\leq R_z(t,z)\cdot z\quad \forall t\in {\mathbb R}\mbox{ and } z\neq 0; \] (R$_2$) $0<\underline{b}=\inf_{t\in{\mathbb R},|z|=1} R(t,z)$; (R$_3$) $|R_z(t,z)|=o(|z|)$ as $z\to 0$ uniformly in $t$; (R$_4$) there exist $0\leq a_1(t)\in L^1({\mathbb R})\cap L^\infty({\mathbb R})$, $\gamma>1$ and $a_2>0$ such that \[ |R_z(t,z)|^\gamma\leq a_1(t)+a_2 R_z(t,z)\cdot z\quad \forall (t,z). \] In [7], Ding showed that (HS) possesses infinitely many homoclinic orbits if, in addition, $H(t,z)$ is even in $z$. The purpose of this paper is to show the same conclusion under a general symmetry condition. Our arguments remain simple even under this general symmetry condition. To state our result, we recall some standard notations concerning group actions of compact subgroups $G$ of the orthogonal group ${\cal O}(2N)$. We let $V$ denote the vector space ${\mathbb R}^{2N}$ considered as a $G$-space. Hence $G$ acts diagonally on $V^k=({\mathbb R}^{2N})^k$, i.e., $g(v^1,\cdots, v^k)=(g v^1,\cdots, gv^k)$ for $g\in G$ and $v^i\in V$ $(k\in {\mathbb N}, i=1,2,\cdots, k)$. If $G$ acts on two subspaces $X$ and $Y$, then a $G$-map $f:X\to Y$ is a continuous map which commutes with the action, i.e., $f(gx)=gf(x)$ for any $g\in G$ and $x\in X$. In the special case where the action on $Y$ is trivial $(gy=y$ for all $g\in G$ and $y\in Y)$ a $G$-map is also called invariant. $A$ subset $A$ of $V^k$ is said to be invariant if $gx\in A$ for every $g\in G$ and $x\in A$. We say that $G$ acts admissibly on $V$ if every $G$-map $\overline{{\cal O}}\to V^{k-1}$, ${\cal O}\subseteq V^k$ an open bounded invariant neighborhood of $0$ in $V^k$, has a zero on $\partial {\cal O}$. Now we can state the symmetry condition. \noindent {\bf (S)} There exists a compact subgroup $G$ of ${\cal O}(2N)$ acting admissibly on $V$ such that $g^t Jg=J$ for every $g\in G$ and $H(t,z)$ is invariant with respect to the action, i.e., $H(t,gz)=H(t,z)$ for all $g\in G$ and $(t,z)\in {\mathbb R}\times {\mathbb R}^{2N}$. Our result reads as follows. \paragraph {Theorem 1.} {\it Let $H$ be of the form {\rm (1.1)} with $L$ satisfying $(L_1)-(L_2)$ and $R$ satisfying $(R_1)-(R_4)$. Suppose, in addition, $H$ satisfies {\rm (S)}. Then {\rm (HS)} possesses infinitely many homoclinic orbits $\{z_k\}$ such that \[ \int_{\mathbb R}[-\frac{1}{2}J\dot{z}_k\cdot z_k-H(t,z_k)]dt\to\infty \mbox{ as } k\to\infty. \]} The Borsuk-Ulam theorem states that $V={\mathbb R}^{2N}$ with the antipodal action of $G=\{I_{2N},-I_{2N}\}$ is admissible. So our result generalizes the result in {\rm [7]}. A simple example of a matrix-valued function satisfying $(L_1)-(L_2)$ is $L(t)=|t|^\theta I_N$ with $\theta>1$, which arises in the study of generalized harmonic oscillator problems. Consider the functions of the form $R(t,z)=b(t)W(z)$, where $b(t)\in C({\mathbb R},{\mathbb R})$, there exist positive constants $\underline{b}\leq\overline{b}$ such that $\underline{b}\leq b(t)\leq\overline{b}$ for all $t\in{\mathbb R}$, and for some integer $m>0$, $W(z)=\sum^m_{i=1}c_i|z|^{\mu i}$ with $c_i>0$ $(1\leq i\leq m)$ and $1<\mu_1\leq \mu_2\leq\cdots\leq\mu_m$. If $\mu>2$, then $R(t,z)$ satisfies $(R_1)-(R_4)$. The preliminary results are in Sec.2, and in Sec.3 is the proof of Theorem 1. \section{Preliminaries} \setcounter{equation}{0} An abstract critical point theorem will be used for proving Theorem 1. This abstract theorem is introduced and proved in [3]. So we shall describe it briefly. For details see [3]. Let $E$ be a Hilbert space with an orthogonal action of a compact Lie group $G$. We are concerned with critical points of an invariant functional $I\in C^1(E,{\mathbb R})$. We need the following assumptions: (A$_1$) There exists an admissible representation $V$ of $G$ such that $E=\oplus_{j\in{\mathbb Z}} E^j$ is a $G$-Hilbert space with $E^j\cong V$ as a representation of $G$ for every $j\in{\mathbb Z}$ (note that ${\mathbb Z}$ can be replaced by ${\mathbb Z}^\ast={\mathbb Z}\backslash\{0\}$, depending on situations). (A$_2$) There exists $a\in {\mathbb R}$ such that for each $k\geq 1$ \[ \inf_{R>0}\sup_{u\in E_k,\| u\| \geq R} I(u)=\lim_{R\to+\infty}\sup_{u\in E_k,\| u\|\geq R}I(u)0}\inf_{u\in E^\bot_{k-1},\| u\|=r}I(u)\to\infty$ as $k\to\infty$. (A$_4$) $d_k=\sup_{u\in E_k}I(u)<\infty$. (A$_5$) Every sequence $u_n\in F_n=E^\bot_{-n-1}=\oplus_{j\geq -n}E^j$ such that $I(u_n)\geq a$ is bounded and $(I|_{F_n})'(u_n)\to 0$ as $n\to\infty$, contains a subsequence which converges in $E$ to a critical point of $I$, which is the so-called (PS)$^\ast$ condition.\\ Now we can state the abstract theorem. \paragraph {Theorem 2.1.} {\it Let $E$ be a $G$-Hilbert space and $I\in C^1(E,{\mathbb R})$ be a $G$-invariant functional satisfying $(A_1)-(A_5)$. Then $I$ has an unbounded sequence of critical values. In fact, for each $k\geq 1$ with $b_k>a$ there exists a critical value $c_k\in [b_k,d_k]$.} \paragraph {Remark 2.2:} In [7], an abstract critical point proposition for even functionals is posed to prove its main result. The proposition requires $I$ to satisfy both (PS)$^\ast$ and (PS)$^{\ast\ast}$ conditions. \paragraph {Remark 2.3:} The above conditions (A$_2$) and (A$_3$) show that the behavior of $I$ is quite interesting. Intuitively, $I$ behaves like fountain (see [3]). Next we consider the symmetric matrix-valued functions $M\in C({\mathbb R},{\mathbb R}^{2N\times 2N})$ of the form \[ M(t)=\left(\begin{array}{lll} 0 & L(t)\\ L(t) & 0 \end{array}\right). \] Suppose that $L$ satisfies (L$_1$) and (L$_2$). Let $A$ be the selfadjoint operator $-J\frac{d}{dt}+M$ with the domain $D(A)\subseteq L^2\equiv L^2({\mathbb R},{\mathbb R}^{2N})$, defined as a sum of quadratic forms. Let $\{E(\lambda)|-\infty<\lambda<+\infty\}$ be the resolution of $A$, and $U=I-E(0)-E(-0)$. Then $U$ commutes with $A$, $|A|$ and $|A|^{1/2}$, and $A=|A|U$ is the polar decomposition of $A$ (see [10]). $D(A)=D(|A|)=D(I+|A|)$ is a Hilbert space equipped with the norm \[ \| z\|_1=\|(I+|A|)z\|_{L^2}\mbox{ for all } z\in D(A), \] where $\|\cdot\| _{L^2}$ is the norm of $L^2$. It is not hard to check that $D(A)$ is continuously embedded in $W^{1,2}\equiv W^{1,2}({\mathbb R},{\mathbb R}^{2N})$ (see [9]). Moreover we have \paragraph {Lemma 2.4:} {\it Suppose $L$ satisfies {\rm (}L$_1${\rm )} and {\rm (}L$_2${\rm )}. Then $D(A)$ is compactly embedded in $L^2$.} For the proof of the above lemma, see [9, Lemma 2.1]. \paragraph {Remark 2.5:} From Lemma 2.4, it is clear that $(I+|A|)^{-1}:L^2\to L^2$ is a compact linear operator. Therefore a standard argument shows that $\sigma(A)$, the spectrum of $A$, consists of eigenvalues numbered by (counted in their multiplicities): \[ \cdots\leq \lambda_{-2}\leq\lambda_{-1}\leq 0<\lambda_1\leq\lambda_2\leq\cdots \] with $\lambda_{\pm k}\to\pm\infty$ as $k\to\infty$, and a corresponding system of eigenfunctions $\{e_k\}_{k\in {\mathbb Z}^\ast}$ of $A$ forms an orthonormal basis in $L^2$ (for the situation here, we use ${\mathbb Z}^\ast$, instead of ${\mathbb Z}$). Now we set $E=D(|A|^{1/2})=D((I+|A|)^{1/2})$. $E$ is a Hilbert space under the inner product \[ (z_1,z_2)_0=(|A|^{1/2}z_1,|A|^{1/2}z_2)_{L^2}+(z_1,z_2)_{L^2} \] and norm \[ \| z\|_0=(z,z)^{1/2}_0=\| (I+|A|)^{1/2}z\|_{L^2}, \] where $(\cdot,\cdot)_{L^2}$ denotes the $L^2$ inner product. Let $E^0=$ ker $A$ (note dim$E^0<\infty$, by Lemma 2.4), $E^+=$Cl$_E$ (span $\{e_1,e_2,\cdots\})$ and $E^-=(E^0\oplus E^+)^{\bot_E}$, where Cl$_ES$ denotes the closure of $S$ in $E$ and $S^{\bot_E}$ denotes the orthogonal complementary subspace of $S$ in $E$. Then \begin{equation} E=E^-\oplus E^0\oplus E^+. \end{equation} Since, by Lemma 2.4, $0$ is at most an isolated eigenvalue of $A$, for later convenience we introduce on $E$ the inner product \[ (z_1,z_2)=(|A|^{1/2}z_1,|A|^{1/2}z_2)_{L^2}+(z^0_1,z^0_2)_{L^2} \] for all $z_i=z^-_i+z^0_i+z^+_i\in E^-\oplus E^0\oplus E^+(i=1,2)$, and the norm \begin{equation} \| z\|=(z,z)^{1/2} \end{equation} for all $z\in E$. Clearly, $\|\cdot\|$ is equivalent to $\|\cdot\|_0$. Moreover, $E$ is continuously embedded in $H^{1/2}({\mathbb R},{\mathbb R}^{2N})$, the Sobolev space of fractional order (see [9]). \paragraph {Lemma 2.6:} {\it Suppose $L$ satisfies (L$_1$) and (L$_2$). Then $E$ is compactly embedded in $L^p$ for all $p\in [2,\infty)$.} For the proof of the above lemma, see [9, Lemma 2.2]. Finally we introduce \begin{equation} a(z,x)=(|A|^{1/2}Uz,|A|^{1/2}x)_{L^2} \end{equation} for all $z,x\in E$. The form $a(\cdot,\cdot)$ is the quadratic form associated with $A$. Clearly, for $z\in D(A)$ and $x\in E$ we have \begin{equation} a(z,x)=(Az,x)_{L^2}=\int_{\mathbb R}(-J\dot{z}+M(t)z)\cdot x. \end{equation} Clearly, $E^-, E^0$ and $E^+$ are orthogonal to each other with respect to $a(\cdot,\cdot)$, and furthermore \begin{eqnarray} &a(z,x)=((P^+-P^-)z,x) &\mbox{ for } z,x\in E\,,\nonumber\\ &a(z,z)=\| z^+\|^2-\| z^-\| ^2 &\mbox{ for } z\in E, \end{eqnarray} where $P^\pm:E\to E^\pm$ are the orthogonal projectors and $z=z^-+z^0+z^+\in E^-\oplus E^0\oplus E^+$. \section{Proof of Theorem 1} \setcounter{equation}{0} Throughout this section, let the assumptions of Theorem 1 be satisfied. Let $E=D(|A|^{1/2})$ with norm (2.2). By (R$_1$) and (R$_2$) we have \begin{equation} R(t,z)\geq \underline{b}|z|^\mu\quad \forall t\in{\mathbb R}\quad\mbox{ and }\quad |z|\geq 1. \end{equation} Also by (R$_4$) and (3.1) we have \begin{equation} |R_z(t,z)|\leq C(1+|z|^{\gamma'-1})\quad \forall (t,z), \end{equation} where $\gamma'=\frac{\gamma}{\gamma-1}$, which, together with (R$_3$), yields that for any $\varepsilon>0$ there is $C_\epsilon>0$ such that \begin{equation} |R_z(t,z)|\leq \varepsilon|z|+C_\varepsilon|z|^{\gamma'-1}\,\,\, \forall (t,z), \end{equation} and \begin{equation} |R(t,z)|\leq \varepsilon|z|^2+C_\varepsilon|z|^{\gamma'}\,\,\, \forall (t,z). \end{equation} Subsequently, $C$ and $C_i$ stand for generic positive constants, not depending on $t$ and $z$. \\ Note that (3.1) and (3.4) imply $\gamma'\geq \mu>2$. Let \[ \varphi(z)=\int_{\mathbb R}R(t,z)\quad \forall z\in E. \] Equations (3.1)-(3.4) imply that $\varphi$ is well-defined, $\varphi\in C^1(E,{\mathbb R})$, and \begin{equation} \varphi'(z)x=\int_{\mathbb R}R_z(t,z)x \quad \forall x,z\in E \end{equation} by Lemma 2.6. In addition, $\varphi'$ is a compact map. To see this, let $z_n\to z$ weakly in $E$. By Lemma 2.6 we can assume that $z_n\to z$ strongly in $L^p$ for $p\in [2,\infty)$. By (3.5) \[ \| \varphi'(z_n)-\varphi'(z)\|=\sup_{\| x\|=1}| \int_{\mathbb R}(R_z(t,z_n)-R_z(t,z))x|. \] By (3.3) and the H$\ddot{\rm o}$lder inequality, for any $R>0$ \begin{eqnarray} \lefteqn{ |\int_{|t|\geq R} (R_z(t,z_n)-R_z(t,z))x| }\nonumber\\ &\leq&C\int_{|t|\geq R}(|z_n|+|z|+|z_n|^{\gamma'-1}+|z|^{\gamma'-1})|x|\\ &\leq&C[\| x\|_{L^2}(\int_{|t|\geq R}|z_n|^2+|z|^2)^{1/2}+ \| x\|_{L^{\gamma'}}(\int_{|t|\geq R}|z_n|^{\gamma'}+|z|^{\gamma'}) ^{(\gamma'-1)/\gamma'}].\nonumber \end{eqnarray} For $\varepsilon>0$, by (3.6) we can take $R_0$ so large that \begin{equation} |\int_{|t|\geq R_0}(R_z(t,z_n)-R_z(t,z))x|<\varepsilon/2 \end{equation} for all $\| x\|=1$ and $n\in{\mathbb N}$. On the other hand, it is well-known (see [13]) that since $z_n\to z$ strongly in $L^2$, \[ \| R_z(\cdot,z_n)-R_z(\cdot,z)\|_{L^2(B_{R_0})}\to 0 \] as $n\to\infty$, where $B_{R_0}=(-R_0,R_0)$. Therefore, there is $n_0\in {\mathbb N}$ such that \begin{equation} |\int_{|t|\leq R_0}(R_z(t,z_n)-R_z(t,z))x|<\varepsilon/2 \end{equation} for all $\| x\|=1$ and $n\geq n_0$. Combining (3.7) and (3.8) yields \[ \| \varphi'(z_n)-\varphi'(z)\|<\varepsilon \quad \forall n\geq n_0. \] Hence $\varphi'$ is compact. Let $a(\cdot,\cdot)$ be the quadratic form given by (2.3), and define \[ I(z)=\frac{1}{2}a(z,z)-\varphi(z) \quad \forall z\in E. \] By (2.5) \[ I(z)=\frac{1}{2}({\| {z^+}\|^2}-{\| {z^-}\|^2})-\varphi(z) \quad \forall z\in E \] for all $z=z^-+z^0+z^+\in E^-\oplus E^0\oplus E^+$. Then $I\in C^1(E,{\mathbb R})$. Note that by (2.4) a standard argument can show that the nontrivial critical points of $I$ on $E$ are homoclinic orbits of (HS). Let $\hat{E}_1=E^-\oplus E^0$ and $\hat{E}_2=E^+$ with $\{e_{-n}\}^\infty_{n=1}$ and $\{e_n\}^\infty_{n=1}$ respectively, where $\{e_n\}_{n\in {\mathbb Z}^\ast}$ is the system of eigenfunctions of $A$ (see Remark 2.5). Then $$E=\hat{E}_1\oplus \hat{E}_2=\oplus_{j\in{\mathbb Z^\ast}}E^j\,,$$ where $E^1=$span$\{e_1,e_2,\cdots,e_{2N}\}$, $E^2=$span$\{e_{2N+1},\cdots,e_{4N}\},\cdots$; \\ $E^{-1}=$span$\{e_{-1},e_{-2},\cdots,e_{-2N}\}$, $E^{-2}=$span$\{e_{-2N-1},\cdots,e_{-4N}\},\cdots$. Set also $E_n=\oplus_{j\leq n}E^j$ and $F_n=E^\bot_{-n-1}=\oplus_{j\geq -n}E^j$ for $j,n\in {\mathbb Z}^\ast$. It remains to check the assumptions of Theorem 2.1. The action of $G$ on $E$ is simply given by $(gz)(t)=gz(t)$. Since $g$ commutes with $J$ and $H(t,z)$ is invariant with respect to the action, it is clear that (A$_1$) is satisfied. Assumption (A$_2$) follows from \paragraph {Lemma 3.1.} {\it For each $k\geq 1$ there exists $R_k>0$ such that $I(z)<0$ for all $z\in E_k$ with $\| z\|\geq R_k$.} \paragraph {Proof.} By (3.4), (R$_1$) and the fact that $|z|^\mu\leq |z|^2$ for $|z|\leq 1$, we have for any $\varepsilon$ with $0<\varepsilon\leq\underline{b}$, \begin{equation} R(t,z)\geq\varepsilon(|z|^\mu-|z|^2)\quad \forall(t,z). \end{equation} Let $d>0$ be such that $\| z\|^2_{L^2}\leq d\| z\|^2$ for all $z\in E$ (by Lemma 2.6) and take $\varepsilon=\min\{\frac{1}{4d},\underline{b}\}$. Then by (3.9) for $z=z^-+z^0+z^+\in E_k$ we have \begin{eqnarray} I(z)&=&\frac{1}{2}{{\| z^+}\|^2}-\frac{1}{2}{{\| z^-}\|^2}- \int_{\mathbb R}R(t,z)\nonumber\\ &\leq&\frac{1}{2}{{\| z^+}\|^2}-\frac{1}{2}{{\| z^-}\|^2}+ \varepsilon\| z\|^2_{L^2}-\varepsilon\| z\|^\mu_{L^\mu}\\ &\leq&{{\| z^+}\|^2}-\frac{1}{4}{{\| z^-}\|^2}+ \frac{1}{4}{{\| z^0}\|^2}-\varepsilon\| z\|^\mu_{L^\mu}.\nonumber \end{eqnarray} Since dim$[E^0\oplus(\oplus_{00$ depend on $k$ but not on $z\in E_k$. Equations (3.10) and (3.11) imply \begin{equation} I(z)\leq\| z^0+z^+\|^2-\frac{1}{4}\| z^-\|^2-\varepsilon C''(k)\| z^0+z^+\| ^\mu \end{equation} for all $z\in E_k$. Equation (3.12) implies that there is $R_k>0$ such that \[ I(z)<0 \quad \forall z\in E_k \mbox{ with }\| z\| \geq R_k. \] \hfill $\diamondsuit$ \medskip Note that the above estimate (3.12) also gives $\sup_{z\in E_k}I(z)<\infty$, that is, (A$_4$) holds. Next, (A$_3$) is a consequence of \paragraph {Lemma 3.2.} {\it There are $r_k>0$, $a_k>0$ $(k\geq 1)$ with $a_k\to\infty$ as $k\to\infty$ such that} \[ I(z)\geq a_k\quad \forall z\in E^\bot_{k-1} \mbox{ with } \| z\| =r_k. \] \medskip \paragraph{Proof.} Define \[ \eta_k=\sup_{z\in E^\bot_{k}\backslash\{0\}}\frac{\| z\|_{L^{\gamma'}}} {\| z\|}. \] Clearly $\eta_k\geq \eta_{k+1}>0$. We claim that \begin{equation} \eta_k\to 0 \quad\mbox{ as }\quad k\to\infty. \end{equation} Suppose $\eta_k\to\eta >0$. Then there is a sequence $z_k\in E^\bot_k$ with $\| z_k\|=1$ and $\| z_k\|_{L^{\gamma^1}}\geq\frac{\eta}{2}$. since $(z_k,e_n)\to 0$ as $k\to\infty$ for each $e_n$ $(n\in {\mathbb Z}^\ast)$, $z_k\to 0$ weakly in $E$ and by Lemma 2.6, $\| z_k\|_{L^{\gamma'}}\to 0$, a contradiction. The claim (3.13) is proved. By (3.4) with $\varepsilon=\frac{1}{4d}$ ($d$ as in the proof of Lemma 3.1) and $C=C_\varepsilon$ we have, for $z\in E^\bot_{k-1}$ \begin{eqnarray*} I(z)&=&\frac{1}{2}\| z\|^2-\int_{\mathbb R}R(t,z)\\ &\geq&\frac{1}{4}\| z\|^2-C\| z\|^{\gamma'}_{L^{\gamma'}}\\ &\geq&\frac{1}{4}\| z\|^2-C\eta^{\gamma'}_{k-1}\| z\|^{\gamma'}. \end{eqnarray*} Taking $r_k=(2\gamma' C\eta^{\gamma'}_{k-1})^{\frac{-1}{{\gamma'}-2}} $ and $a_k=(\frac{1}{4}-\frac{1}{2\gamma'})r^2_k$ \,\, one obtains \[ I(z)\geq a_k\quad \forall z\in E^\bot_{k-1}\mbox{ with } \| z\| =r_k. \] Since ${\gamma'}>2$, equation (3.13) shows that $a_k\to\infty$ as $k\to\infty$. \hfill $\diamondsuit$ \paragraph {Lemma 3.3} {\it $I$ satisfies (A$_5$).} \paragraph {Proof.} Let $I_n=I|_{F_n}$. Suppose $z_n\in F_n$ such that $0\leq I(z_n)\leq C$ and $\varepsilon_n=\| I'_n(z_n)\|\to 0$. By definition and (R$_1$) \begin{eqnarray} I(z_n)-\frac{1}{2}I'_n(z_n)z_n&=&\int_{\mathbb R}(\frac{1}{2}R_z(t,z_n)z_n-R(t,z_n))\nonumber\\ &\geq&(\frac{1}{2}-\frac{1}{\mu})\int_{\mathbb R}R_z(t,z_n)z_n\nonumber\\ &\geq&(\frac{\mu}{2}-1)\int_{\mathbb R}R(t,z_n). \end{eqnarray} Equation (3.14) and hypothesis (R$_4$) give $\| R_z(t,z_n)\|^\gamma_{L^\gamma}\leq C(1+\| z_n\|)$, and hence by Lemma 2.6, \begin{eqnarray*} \| z^+_n\|^2&=&I'(z_n)z^+_n+\int_{\mathbb R}R_z(t,z_n)z^+_n\\ &\leq&C\| z^+_n\| (1+\| R_z(t,z_n)\|_{L^\gamma}). \end{eqnarray*} Thus \begin{equation} \| z^+_n\|\leq C(1+\| z_n\|^{1/\gamma}). \end{equation} Similarly we have \begin{equation} \| z^-_n\|\leq C(1+\| z_n\|^{1/\gamma}). \end{equation} If $E^0=\{0\}$, (3.15) and (3.16) imply $\| z_n\| \leq$ Const $\forall n$. Suppose $E^0\neq\{0\}$. For $z\in E$, let \begin{eqnarray*} z^1(t)=\left\{\begin{array}{lllll} z(t) & \mbox{ if } &|z(t)|<1, \\ 0 & \mbox{ if } & |z(t)|\geq 1, \end{array}\right. z^2(t)=\left\{\begin{array}{llll} 0 & \mbox{ if } &|z(t)|<1, \\ z(t) & \mbox{ if } & |z(t)|\geq 1.\end{array}\right. \end{eqnarray*} Since by Lemma 2.6 \[ \int_{\mathbb R}|z^1_n|^\mu\leq\int_{\mathbb R}|z^1_n|^2\leq\int_{\mathbb R}|z_n|^2\leq C\| z_n\| ^2, \] we have \begin{equation} \| z^1_n\|_{L^\mu}\leq C\| z_n\|^{2/\mu}. \end{equation} By (3.1) and (3.14), \begin{equation} \| z^2_n\|_{L^\mu}\leq C(1+\| z_n\|^{1/\mu}). \end{equation} By $L^2$ orthogonality and H$\ddot{\rm o}$lder's inequality with $\mu'=\frac{\mu}{\mu-1}$, \begin{eqnarray*} \| z^0_n\|^2_{L^2}&=&(z^0_n,z_n)_{L^2}\\ &\leq&\| z_n^0\|_{L^{\mu'}}(\| z^1_n\|_{L^\mu}+ \| z^2_n\|_{L^\mu}). \end{eqnarray*} Hence since dim$E^0<\infty$ and (3.17)-(3.18) hold, one sees \begin{equation} \| z^0_n\|_{L^\mu}\leq C(\| z_n\|^{2/\mu}+ \| z_n\|^{1/\mu}). \end{equation} The combination of (3.15)-(3.16) and (3.19) shows that again $\| z_n\|\leq$ Const. Finally since $\varphi'$ is compact, a standard argument shows that $\{z_n\}$ has a convergent subsequence. \hfill $\diamondsuit$ \medskip {\bf Proof of Theorem 1.} What we have done so far shows that $I$ satisfies all the assumptions of Theorem 2.1. Hence $I$ has a positive critical value sequence $\{c_k\}$ with $c_k\to\infty$. Let $z_k$ be the critical point of $I$ such that $I(z_k)=c_k$. Then $z_k$ is a homoclinic orbit of (HS) and \begin{eqnarray*} \int_{\mathbb R}(-\frac{1}{2}J\dot{z}_k\cdot z_k-H(t,z_k))dt=I(z_k)=c_k\to\infty \end{eqnarray*} as $k\to\infty$. \hfill $\diamondsuit$ \paragraph{Acknowledgments.} The author would like to thank Professor Yangheng Ding, Academia Sinica of P. R. China, for his suggestions. \begin{thebibliography}{00} \bibitem{key1} Ambrosetti, A. \& Rabinowitz, P. H., {\it Dual variational methods in critical point theory and applications}, J. Funct. Anal., 14 (1973), 349-381. \bibitem{key2} Ambrosetti, A. \& Coti-Zelati, V., {\it Multiple homoclinic orbits for a class of conservative systems}, C. R. Acad. Sci. Paris, 314 (1992), 601-604. \bibitem{key3} Bartsch, T. \& Willem, M., {\it Periodic solutions of non-autonomous Hamiltonian systems with symmetries}, J. Reine. Angew. Math. 451 (1994), 149-159. \bibitem{key4} Coti-Zelati, V., Ekland, I. \& S$\acute{\rm e}$r$\acute{\rm e}$, E., {\it A variational approach to homoclinic orbits in Hamiltonian systems}, Math. Ann. 288 (1990), 133-160. \bibitem{key5} Coti-Zelati, V. \& Rabinowitz, P. H. {\it Homoclinic orbits for a second order Hamiltonian systems possessing superquadratic potentials}, J. Am. Math. Soc., 4 (1991), 693-727. \bibitem{key6} Ding, Y., {\it Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems}, Nonlinear Analysis T.M.A., 25 (1995), 1095-1113. \bibitem{key7} Ding, Y., {\it Infinitely many Homoclinic orbits for a class of Hamiltonian systems with symmetry}, Chin. Ann. of Math. 19B, 22 (1998), 167-178. \bibitem{key8} Ding, Y. \& Giradi, M., {\it Periodic and homoclinic solutions to a class of Hamiltonian systems with potentials changing sign}, Dyn. Sys. and Appl., 2 (1993), 131-145. \bibitem{key9} Ding, Y. \& Li, S., {\it Homoclinic orbits for the first-order Hamiltonian systems}, J. Math. Anal. and Appl., 189 (1995), 585-601. \bibitem{key10} Edmunds, D. E. \& Evans, W. D., {\it Spectral theory and differential operators}, Clarenden Press, Oxford, 1987. \bibitem{key11} Hofer, H. \& Wysocki, K., {\it First-order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems}, Math. Ann., 288 (1990), 483-503. \bibitem{key12} Omana, W. \& Willem, M., {\it Homoclinic orbits for a class of Hamiltonian systems}, Diff. and Int. Eq., 5 (1992), 1115-1120. \bibitem{key13} Rabinowitz, P. H., {\it Minimax methods in critical point theory with applications to differential equations}, CBMS Reg. Conf. Ser. in Math., 65, A.M.S. Providence, 1986. \bibitem{key14} Rabinowitz, P. H., {\it Homoclinic orbits for a class of Hamiltonian systems}, Proc. Royal. Soc. Edinburgh, 114A (1990), 33-38. \bibitem{key15} Rabinowitz, P. H., \& Tanaka, K., {\it Some results on connecting orbits for a class of Hamiltonian systems}, Math. Z., 206 (1991), 473-499. \bibitem{key16} S$\acute{\rm e}$r$\acute{\rm e}$, E., {\it Existence of infinitely many homoclinic orbits in Hamiltonian systems}, Math. Z., 209 (1992), 27-42. \bibitem{key17} S$\acute{\rm e}$r$\acute{\rm e}$, E., {\it Looking for the Bernoulli shift}, Ann. Inst. H. Poincar$\acute{\rm e}$ Anal. Non. Lineaire 10, no. 5, (1993), 561-590. \bibitem{key18} Tanaka, K., {\it Homoclinic orbits in a first-order superquadratic Hamiltonian system: convergence of subharmonic orbits}, J. Diff. Eq., 94 (1991), 315-339. \end{thebibliography} \medskip \noindent {\sc Cheng Lee }\\ Department of Mathematics \\ National Changhua University of Education \\ Changhua, Taiwan. 50058 R.O.C. \\ e-mail address: clee@math.ncue.edu.tw \end{document}