\magnification = \magstephalf \hsize=14truecm \hoffset=1truecm \parskip=5pt \nopagenumbers \input amssym.def \font\eightrm=cmr8 \font\eighti=cmti8 \font\eightbf=cmbx8 \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--1995/12\hfil Multibump solutions \hfil\folio} \def\leftheadline{\folio\hfil Paul H. Rabinowitz \hfil EJDE--1995/12} \voffset=2\baselineskip \vbox {\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations, Vol. {\eightbf 1995}(1995) No. 12, pp. 1--21.\hfill\break ISSN 1072-6691: URL: http://ejde.math.swt.edu (147.26.103.110)\hfil\break telnet (login: ejde), ftp, and gopher access: ejde.math.swt.edu or ejde.math.unt.edu} \footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt % 1991 {\eighti Subject Classification:} 34C37, 49M10, 58E99, 58F05. \hfil\break {\eighti Key words and phrases:} homoclinic, multibump solution, calculus of variations. \hfil\break \copyright 1995 Southwest Texas State University and University of North Texas.\hfil\break Submitted July 25, 1995. Published September 15, 1995.\hfil\break Supported by NSF grant DAAL03-87-12-0043, and by U. S. Army contract DAAL03-87-12-0043} } \bigskip\bigskip \centerline{MULTIBUMP SOLUTIONS FOR AN ALMOST PERIODICALLY} \centerline{FORCED SINGULAR HAMILTONIAN SYSTEM} \smallskip \centerline{Paul H. Rabinowitz} \bigskip\bigskip {\eightrm\baselineskip=10pt \narrower \centerline{\eightbf Abstract} This paper uses variational methods to establish the existence of so-called multibump homoclinic solutions for a family of singular Hamiltonian systems in $\Bbb R^2$ which are subjected to almost periodic forcing in time. \bigskip} \bigskip \def\C{{\underline C}} \def\RR{{\cal R}} \def\HH{{\cal H}} \def\LL{{\cal L}} \def\NN{{\cal N}} \def\BB{{\cal B}} \def\VV{{\cal V}} \def\KK{{\cal K}} \def\pxo{{ (X_0) }} \def\olim{\mathop{\overline{\rm lim}}} \def\ulim{\mathop{\underline{\rm lim}}} \def\tv{{\|\!\!\|\!\!|}} \def\loc{\rm loc} \overfullrule=0pt \centerline{\bf Introduction} \medskip This paper is a sequel to [1] where the existence of homoclinic solutions was proved for a family of singular Hamiltonian systems which were subjected to almost periodic forcing. More precisely, consider the Hamiltonian system $$ \ddot q + a (t) W' (q) = 0 \leqno(\hbox{HS}) $$ where $a$ and $W$ satisfy {\narrower\smallskip \item{($a_1$)} $a(t)$ is a continuous almost periodic function of $t$ with $a(t)\ge a_0>0$ for all $t\in \Bbb R$. \smallskip \item{($W_1$)} There is a $\xi\in\Bbb R^2\backslash\{ 0\}$ such that $W\in C^2(\Bbb R^2\backslash \{\xi\}, \Bbb R)$. \smallskip \item{($W_2$)} $\displaystyle{\lim_{x\to\xi}W(x)=-\infty}$. \smallskip \item{($W_3$)} There is a neighborhood $\NN$ of $\xi$ and $U\in C^1(\NN\backslash\{\xi \}, \Bbb R)$ such that $|U(x)|\to\infty$ as $x\to\xi$ and $$ |U'(x)|^2\le -W(x)\quad\hbox{for $x\in\NN\backslash\{\xi\}$}, $$ \item{($W_4$)} $W(x) 0 \} . $$ The main results in [1] were that $I$ possesses infinitely many critical points in $\Gamma^+$ and $\Gamma^-$ with corresponding critical values near $$ c^\pm = {\mathop {\rm inf}\limits_{\Gamma^\pm} } I \leqno (0.3) $$ Moreover if $c^\pm$ is attained by $I$ at $Q^\pm\in \Gamma^\pm$ with $Q^\pm$ an isolated critical point of $I$, then there is an unbounded sequence $(\sigma_m ) \subset \Bbb R$ such that $I$ has a local minimum near $Q^\pm (t-\sigma_m)$ for large $m$. The numbers $(\sigma_m)$ stem from the almost periodicity of the function $a(t)$ which implies there is such a sequence satisfying $$ \| a (\cdot ) - a(\cdot + \sigma_m) \|_{L^\infty (\Bbb R )} \rightarrow 0 \leqno (0.4) $$ as $m \rightarrow \infty$. We do not know whether $c^+$ (resp.\ $c^-$) is attained by $I\equiv I_a$ for the given almost periodic function $a(t)$. However there is always an $\alpha$ in $\HH (a)$, the hull of $a$, i.e.\ the $L^\infty$ closure of the set of translates of $a(t)$ for which the infimum is achieved by the corresponding $I_\alpha$. The main goal of the current paper is to show that when there is an isolated minimizer $Q^\pm \in \Gamma^\pm$ of $I$ with $I(Q^\pm ) = c^\pm$, then (HS) possesses so called multibump solutions. To state this a bit more precisely, for $s \in \Bbb R$ and $q \in E$, set $$ \tau_s q(t) = q(t-s) .\leqno (0.5) $$ We will prove that for any $k \in \Bbb N$, near $\sum^k_1 \tau_{\sigma_{j_i}} Q^\pm$, there is an actual homoclinic solution of (HS) provided that e.g. $0<\sigma_{j_1} < \cdots < \sigma_{j_k}$ and $\sigma_{j_1}, \sigma_{j_{i+1}} - \sigma_{j_i}$ are sufficiently large. If both $Q^+$ and $Q^-$ are isolated minimizers, there is a more complicated existence statement. The requirement that $Q^+$ be isolated is the analogue for the variational approach taken here of the related assumption that one has a transversal intersection of stable and unstable manifolds for a Poincar\'e map associated with (HS) at a homoclinic point corresponding to $Q^+$. An exact formulation of the main existence theorem and its proof will be given in \S1. Some extensions and related results will be carried out in \S2. Various technical results required for the proofs will be treated in \S3. There have been several recent papers, beginning with S\'er\'e [2], which use methods from the calculus of variations to get the existence of multibump homoclinic or heteroclinic solutions of Hamiltonian systems. See e.g. Bessi [3], Bolotin [4], Caldiroli and Montecchiari [5], Coti Zelati and Rabinowitz [6-7], Giannoni and Rabinowitz [8], Montecchiari and Nolasco [9], Rabinowitz [10-11], S\'er\'e [12], and Strobel [13]. Aside from [9] these papers deal with periodically forced Hamiltonian systems. Reference [9] treats a perturbation with arbitrary time dependence of a time periodically forced potential which is a superquadratic function of $q$. See also [11] in this regard. Recently Buffoni and S\'er\'e [14] found multibump solutions for an autonomous superquadratic Hamiltonian system. Our work in [1] was motivated in part by [15] and [11] -- see also Tanaka [16] - where the existence of basic homoclinic and multibump solutions was studied for (HS) under periodic forcing and weaker conditions than ($a_1$), ($W_1$) - (W$_5$). The second major influence on [1] was the recent work of Serra, Tarallo and Terracini [17] who found a basic homoclinic solution for a family of superquadratic Hamiltonian systems under almost periodic forcing. See also Bertotti and Bolotin [18]. Some results on multibump homoclinics in the setting of [17] have been obtained by Spradlin [19] and as we recently learned by Coti Zelati, Montecchiari and Nolasco [20]. Lastly there has been some recent work in the setting of [15] by Caldiroli and Nolasco [21] who study an autonomous problem and under additional hypotheses on the potential find basic homoclinics which wind $k$ times around the singularity for any $k \in \Bbb N$. \bigskip \centerline {\bf \S1. Multibump solutions} \medskip The existence of multibump solutions of (HS) will be studied in this section. In order to formulate the main result, some preliminaries and notations are needed. Let $B_r(t)$ denote an open ball of radius $r$ about $x \in E$. As was noted in the Introduction, by $(a_1)$, there is an unbounded sequence $(\sigma_m) \subset \Bbb R$ such that $$ \| \tau_{-\sigma_m} a - a\|_{L^\infty} \rightarrow 0 \leqno (1.1) $$ as $m \rightarrow \infty$. Fix $k \in \Bbb N$ and let $\sigma_{j_1} < \cdots < \sigma_{j_k}$ with $\sigma_{\sigma_1}, \ldots, \sigma_{j_k} \in (\sigma_m)$. Set $\beta_0 = - \infty$, $\beta_k = \infty$, and for $1 \leq i \leq k - 1, \beta_i = {1\over 2} (\sigma_{j_i} + \sigma_{j_{i+1} } )$. For $x \in E$, set $$ \tv x \tv = \max_{1\leq i \leq k} \| x\|_{W^{1,2} [\beta_{i-1}, \beta_i]}. $$ Thus $\tv x \tv$ is an equivalent norm on $E$. Let $\BB_r (x)$ denote the open ball of radius $r$ about $x \in E$ under $\tv \cdot \tv$. Let $$ \KK = \{ q \in E \backslash \{ 0 \} | I' (q) = 0 \} , $$ i.e., ${\cal K}$ is the set of nontrivial critical points of $I$ or equivalently solutions of (HS) that are homolinic to $0$. Our main result can now be stated: \proclaim{Theorem 1.2}. Let $(a_1), (W_1) - (W_5)$ be satisfied. Suppose that $c^\pm$ (in (0.3)) is attained at an isolated critical point $Q(\in \Gamma^+)$. Let $k \in \Bbb N$ and $\sigma_{j_1} < \cdots <\sigma_{j_k} \in (\sigma_m)$. Set $Q_k = \sum^k_1 \tau_{\sigma_{j_i} } Q$. Then there is an $r_0 > 0$ and an $\ell = \ell (r)$ defined for $0 < r < r_0$ such that whenever $\sigma_{j_1} \geq \ell, \sigma_{j_{i+1}} - \sigma_{j_i} \geq \ell, 1 \leq i \leq k - 1$, $I$ possesses a local minimum in ${\cal B}_r (Q_k)$. \noindent{\bf Remark 1.3}. $\ell$ is independent of $k$. As will be seen in \S2, this leads to the existence of infinite bump solutions of (HS) via a simple limit process. In order to prove Theorem 1.2, some technical preliminaries are required. They will be stated next and their proofs will be given in \S3. The first provides a lower bound for $\| I'\|$ in an annular neighborhood of an isolated minimizer of $I$. \proclaim{Proposition 1.4}. Let $(a_1), (W_1) - (W_5)$ be satisfied and suppose $Q\in \Gamma^+$ is an isolated critical point of $I$ with $I(Q) = c^+$. Then there is an $r_1 > 0$ and $\delta = \delta (r,\underline r)$ defined for $0 < \underline r < r\le r_1$ such that $\| I'(x)\| \geq 4\delta$ if $x \in \overline B_r (Q) \backslash B_{\underline r} (Q)$. The next result concerns the existence of a vector field that plays an important role in the proof of Theorem 1.2. \proclaim{\bf Proposition 1.5}. Under the hypotheses of Proposition 1.4, there is an $r_2 > 0$, a real valued function $\ell_0 (r,\rho)$ defined for $0 < \rho < r \leq r_2$, and a locally Lipschitz continuous function ${\cal V} : \overline B_r (Q_k) \rightarrow E$ satisfying: $$ \tv {\cal V} (x)\tv \leq 3 ,\leqno (1.6) $$ and $$ I'(x) {\cal V}(x) \geq \delta (6r,{\rho\over 8} )\quad \hbox {for}\quad x \in \overline {\cal B}_r (Q_k) \backslash {\cal B}_\rho (Q_k) \leqno (1.7) $$ provided that $j_1 \geq \ell_0$, $\sigma_{j_{i+1}} - \sigma_{j_i} \geq \ell_0$, $1 \leq 1 \leq k - 1$. Moreover defining $$ \Phi_i (x) \equiv \int^{\beta_i}_{\beta_{i-1}} {\cal L} (x)\, dt \qquad 1 \leq i \leq k ,\leqno (1.8) $$ writing $x = Q_k + z$ where $z \in {\cal B}_r (0)$ and setting $z_i = z|^{\beta_i}_{\beta_{i-1}} ,$ then $$ \Phi'_i (x) {\cal V} (x) \geq \delta (6r, {r\over 8} ) \quad \hbox {if} \quad {r\over 4} \leq \| z_i\|_{W^{1,2}[\beta_{i-1}, \beta_i]} \leq r \leqno (1.9) $$ and $$ \Phi_i' (x) {\cal V} (x) \geq \delta (r, {\rho\over 8} )\quad \hbox {if} \quad \rho \leq \| z_i\|_{W^{1,2}[\beta_{i-1}, \beta_i]} \leq r \leqno (1.10) $$ The next preliminary yields a decay estimate for solutions of (HS) that are homoclinic to $0$: \proclaim{Proposition 1.11}. Let $P \in E$ be a solution of (HS). Then there are constants $\gamma, A, R > 0$ such that $$ |P (t) | + |P' (t) | + |P'' (t)| \leq Ae^{-\gamma |t|} \quad \hbox {for} \quad |t| \geq R. \leqno (1.12) $$ The final simple preliminary concerns the existence of minimizers of $I$ in $\BB_\alpha (Q_k)$. \proclaim{Proposition 1.13}. For any $\alpha>0$, there is a $P = P_\alpha \in \overline \BB_\alpha (Q_k)$ such that\break $I(P) = {\mathop {\rm inf}\limits_{\overline \BB_\alpha (Q_k)} } I$. Before beginning the formal proof of Theorem 1.2, we briefly indicate the strategy of the argument. After a suitable choice of parameters, $r, \ell (r)$, etc. by Proposition 1.5, $I' \not = 0$ in $\overline{\BB}_r (Q_k) \backslash \BB_\rho (Q_k)$. If $I$ does not have a local minimum in $\BB_\rho (Q_k)$, by Proposition 1.13, ${\mathop {\rm inf}\limits_{\BB_\rho (Q_k)}} I$ is attained at $z \in \partial \BB_\rho (Q_k)$. An ordinary differential equation is introduced using the function ${\cal V}$ of Proposition 1.5 and with initial condition $\eta (0) = z$. The solution trajectory $\eta (s)$ lies in $\BB_r (Q_k) \backslash \BB_\rho (Q_k)$ for all $k > 0$ and analyzing the behavior of $\eta$ leads to a contradiction. \bigskip \noindent {\bf Proof of Theorem 1.2}: The proof involves variants of arguments from [7] and [1]. For convenience, set $c = c^+$. Choose $r_0 =\min (r_1,r_2)$ and let $\delta_1 (r) = \delta (r, {r\over 4} )$ as given by Proposition 1.4 and $\ell_1(r)\geq\ell_0(r,{r\over 4})$ as given by Proposition 1.5. Set $$ \epsilon = \epsilon (r) = r \delta_1 (r)/48 . \leqno (1.14) $$ For $\ell_1$ sufficiently large, $$ I(Q_k) \leq k( c + {\epsilon\over 2} ) \leqno (1.15) $$ Indeed, $$ I (Q_k) \leq (I(Q_k) - \sum^k_1 I(\tau_{\sigma_{j_i }} Q)) + \sum^k_1 I (\tau_{\sigma_{j_i }} Q) . \leqno (1.16) $$ By (1.1), $$ \leqalignno { I(\tau_{\sigma_{j_i }} Q) & = \int_\Bbb R ({1\over 2} | \tau_{\sigma_{j_i }} \dot Q |^2 - a(x) W (\tau_{\sigma_{j_i }} Q))\, dt & (1.17) \cr & = I(Q) + \int_\Bbb R (a(x) - \tau_{- \sigma_{j_i }} a(t)) W(Q)\, dt \cr & \leq c + {\epsilon\over 4} , \qquad 1 \leq i \leq k\cr} $$ for $\ell_1$ sufficiently large. Moreover $$ \leqalignno { I(Q_k) & - \sum^k_1 I (\tau_{ \sigma_{j_i }} Q) = 2\int_\Bbb R \sum_{ i,p\atop i\not = p} \tau_{ \sigma_{j_{i }} } \dot Q \cdot \tau_{ \sigma_{j_{p }}} \dot Q\, dt & (1.18)\cr & - \int_\Bbb R a(x) (W(Q_k) - \sum^k_1 W (\tau_{ \sigma_{j_{i} }} Q))\, dt\cr} $$ Writing $$ \int_\Bbb R \tau_{ \sigma_{j_{i }}} \dot Q\cdot \tau_{ \sigma_{j_{p }}} \dot Q\, dt = \sum^k_1 \int^{\beta_i}_{\beta_{i-1}} \tau_{ \sigma_{j_{i }}} \dot Q \cdot \tau_{ \sigma_{j_{p }}} \dot Q\, dt \,, \leqno (1.20) $$ in each interval $[\beta_{i-1}, \beta_i]$, for $\ell_1$ large compared to $R$, at least one factor of the integrand is $\leq e^{-\gamma\ell_1/4}$ via Proposition 1.11. Similarly $$ \leqalignno { \int_\Bbb R & a(x) (W(Q_k) - \sum^k_1 W (\tau_{ \sigma_{j_{i }}} Q))\, dt & (1.21) \cr = & \sum^k_1 \int^{\beta_i}_{\beta_{i-1}} a(x) (W(Q_k) - \sum^k_1 W (\tau_{ \sigma_{j_{i }}} Q ))\, dt\cr} $$ and on $[\beta_{i-1}, \beta_i ],$ $$ | W (Q_k (t)) - W(\tau_{ \sigma_{j_{i }}} Q(t)) | \leq M_1 \sum_{p \not = i} | \tau_{ \sigma_{k_p }} Q(t)| \leqno (1.22) $$ where $M_1$ depends on $L^\infty$ bounds for $W'(P)$ for $P$ near $\tau_{ \sigma_{j_i}} Q$. By (1.22) and Proposition 1.11 again, each integral in (1.21) is exponentially small in $\ell_1$. Hence for $\ell_1$ sufficiently large, (1.15) holds via (1.16) - (1.22). Note that $\ell_1$ is independent of $k$. Similar estimates show $$ c - {\epsilon\over 4} \leq \Phi_i (Q_k) \leq c + {\epsilon\over 4} , \qquad 1 \leq i \leq k \leqno (1.23) $$ Next a family of cutoff functions will be introduced. Let $\psi_i (x), \chi_i (x)$ be locally Lipschitz continuous for $x \in \overline \BB_r (Q_k), 1 \leq i \leq k$ and satisfy $$ \psi_i (x)\;\cases { = 0 & if $ \Phi_i (x) \geq c + 2\epsilon$ \cr = 1 & if $\Phi_i (x) \leq c + \epsilon$\cr \in (0,1)& otherwise.\cr} \leqno (1.24) $$ $$ \chi_i (x)\;\cases{ = 0 & if $\Psi_i (x) \leq c - 2\epsilon$ \cr = 1 & if $\Psi_i (x) \geq c - \epsilon$\cr \in (0,1)& otherwise.\cr} \leqno (1.25) $$ Set $$ \psi (x) = \prod^k_1 \psi_i (x);\quad \chi (x) = \prod^k_1 \chi_i (x) \leqno (1.26) $$ choose $\rho = \rho (r)$ so that $$ 0 < \rho < {\epsilon\over 24} \leqno (1.27) $$ and $$ \sup_{\overline B_{4\rho} (Q)} I(x) \leq c + {\epsilon\over 8} \leqno (1.28) $$ If $I$ has a local minimum in $\BB_\rho ( Q_k )$, the Theorem is proved. Thus suppose this is not the case. Then by Proposition 1.13, there is a $z\in \partial \BB_\rho (Q_k)$ such that $$ I(z) = {\rm inf}_{x \in \overline\BB_\rho (Q_k)} I(x) \leqno (1.29) $$ Consider the ordinary differential equations $$ {d\eta\over ds} = -\psi (\eta ) \chi (\eta ) {\cal V} (\eta) \leqno (1.30) $$ where ${\cal V}$ is given by Proposition 1.5 (with $\rho = \rho (r)$ satisfying (1.27) - (1.28)). Note that by (1.7), $I'(x) \not = 0$ for $x \in \overline \BB_r (Q_k) \backslash \BB_\rho (Q_k)$. As initial conditions for (1.30), take $\eta (0) = z$. Since $$ \Phi_i (z) = \Phi_i (Q_k) + \int^1_0 \Phi'_i ( sQ_k + (1-s)z) (Q_k - z) \, ds , \leqno (1.31) $$ $\| z - Q_k \|_{W^{1,2} [\beta_{i-1}, \beta_i ]} \leq \rho$, and $\Phi'_i $ is bounded in $\BB_\rho (Q_k)$, by making $\rho$ still smaller if necessary, it can be assumed that $$ | \Phi_i (z) - c| \leq {\epsilon\over 2} \leqno (1.32) $$ Therefore $\psi (z) = \chi (z) = 1 .$ The solution of (1.30) certainly exists for small $s > 0$. We claim it exists for all $s > 0$ and lies in $\overline \BB_r (Q_k)$. Otherwise for some $i$, some $s_1 < s_2$, $s_2$ being minimal and all $s \in [s_1, s_2]$, $$ \leqalignno { \| \eta (s_1) &- Q_k \|_{W^{1,2} [\beta_{i-1}, \beta_i] } = {r \over 2} \leq \| \eta (s) - Q_k \|_{W^{1,2} [\beta_{i-1},\beta_i]} & (1.33)\cr \leq & \| \eta (s_2) - Q_k\|_{W^{1,2} [\beta_{i-1}, \beta_i]} = r . \cr} $$ Therefore $$ \leqalignno { {r\over 2} & \leq \| \eta (s_1) - \eta (s_2) \|_{W^{1,2} [\beta_{i-1},\beta_i]} & (1.34)\cr & = \left \| \int^{s_2}_{s_1} {d\eta\over ds} \, ds \right \| \leq \int^{s_2}_{s_1} \psi (\eta (s)) \chi (\eta (s)) \| {\cal V} (\eta (s))\|_{W^{1,2} [\beta_{i-1}, \beta_i]}\,ds \cr & \leq 3 \int^{s_2}_{s_1} \psi (\eta (s))\chi (\eta (s))\, ds \cr} $$ via (1.30) and (1.6). By (1.9) $$ \leqalignno { \Phi_i (\eta (s_1)) - \Phi_i (\eta(s_2)) &= \int^{s_1}_{s_2} \Phi'_i (\eta (s)) {d\eta\over ds}\,ds \cr & = \int^{s_2}_{s_1} \psi (\eta (s)) \chi (\eta (s)) \Phi'_i (\eta (s)) {\cal V} (\eta (s))\,ds & (1.35) \cr &\geq \delta_1(r) \int^{s_2}_{s_1} \psi (\eta (s)) \chi ( \eta (s))\, ds\cr} $$ Combining (1.34)-(1.35) yields $$ 8\epsilon = {r\delta_1\over 6} \leq \Phi_i (\eta (s_1)) - \Phi_i (\eta (s_2)) . \leqno (1.36) $$ Due to the definition of $\psi $ and $\chi$, and the form of (1.30), $$ \Phi_i (\eta (s)) \in (c - 2\epsilon, c + 2\epsilon ) \leqno (1.37) $$ for all $s \in [0,s_2]$. Hence (1.36) is not possible and as claimed $\eta (s)$ lies in $\BB_r (Q_k)$ for all $s > 0$. Next observe that $\eta (s) \not\in \BB_\rho (Q_k)$ for all $s > 0$. Indeed $$ {dI\over ds} (\eta (s)) |_{s=0} = - I' (z) {\cal V} (z) < 0 \leqno (1.38) $$ by (1.7) so $I(\eta (s))$ decreases for small $s$. Thus for such $s$, $I(\eta (s)) < I(z)$ and $\eta (s) \not \in \BB_\rho (Q_k)$ by the choice of $z$. Moreover as long as $\eta (s) \in \overline\BB_r (Q_k)\backslash \BB_\rho (Q_k),$ as in (1.38), $$ {dI\over ds} (\eta (s)) = - \psi (\eta (s)) \chi (s)) I' (\eta (s)) {\VV} (\eta (s)) \leq 0 \leqno (1.39) $$ so $\eta (s)$ can never return to $\overline \BB_\rho (Q_k).$ Suppose for the moment that $$ \psi (\eta (s)) = 1 = \chi (\eta (s)) \leqno (1.40) $$ for all $s > 0$. Then by (1.30) and (1.7) again, $$ \leqalignno { I(\eta (s)) & = I(z) + \int^s_0 I' (\eta (s)) \VV (\eta (s))\, ds & (1.41) \cr & \leq I (z) - \delta (6r,{\rho\over 8} ) s\cr} $$ In particular for large $s$, $$ I(\eta (s)) < 0 . \leqno (1.42) $$ But $I(x) \geq 0 $ for all $x \in E$ so (1.42) cannot occur. Consequently $I$ must have a local minimizer in $\BB_\rho (Q_k)$ and Theorem 1.2 follows. It remains to verify (1.40). If it does not hold, there is a smallest $s^* > 0$ beyond which (1.40) is violated. Thus for some $i$, $ |\Phi_i (\eta (s^*)) - c| = \epsilon$. Suppose $$ \Phi_i (\eta (s^*)) = c - \epsilon . \leqno (1.43) $$ We will show this leads to the construction of a function $P\in \Gamma^+$ with $I(P) < c = c^+$. Hence (1.43) is not possible and $\chi (\eta (s)) \equiv 1 $ for $s > 0$. To find $P$, note first that $$ {1\over 2} \| \eta (s^*) - Q_k \|_{L^\infty [\beta_{i-1},\beta_i] } \leq \|\eta (s^*) - Q_k \|_{W^{1,2} [\beta_{i-1},\beta_i]} \leq r \leq r_0 . \leqno (1.44) $$ Hence $\eta (s^*)$ is close to $Q_k$ in $L^\infty [\beta_{i-1}, \beta_i]$. By estimates as in (1.16) and (1.22) using Proposition 1.11, $Q_k$ is close to $\tau_{\sigma_{j_i}} Q $ in $L^\infty [\beta_{i-1}, \beta_i]$. Hence $WN(\eta (s^* ))|^{\beta_i}_{\beta_{i-1}} )$ is near $WN(\tau_{\sigma_{j_{i}}} Q|^{\beta_i}_{\beta_{i-1} } ) = WN(Q|^{\beta_i - \sigma_{j_i}}_{\beta_{i-1} - \sigma_{j_i} } ) $ and for $\ell_1$ large, this latter quantity is near $WN(Q) = d(Q) > 0$. As was noted earlier, it can be assumed that $\ell_1$ is large compared to $R$ of Proposition 2.11 and in particular $Q$ is exponentially small for $|t| \geq {\ell_1\over 4}$. Hence for $t \in [\beta_{i-1} , \beta_{i-1} + {\ell_1 \over 3} ]$, $Q_k$ satisfies an estimate of the form $$ \leqalignno { |Q_k (t) | &= | \sum^k_1 \tau_{\sigma_{j_i} } Q(t)| \leq \sum^k_1 |Q(t-\sigma_{j_i} )| & (1.45) \cr & \leq A \sum^k_1 e^{-\gamma {\ell_1\over 3} i} \leq 2 A e^{-\gamma {\ell_1\over 3} }\cr} $$ with a similar estimate for $\dot Q_k$. Likewise for $t \in [\beta_i - {\ell_1\over 3} , \beta_i ]$, $$ |\dot Q_k (t) | + |Q_k(t) | \leq 2Ae^{-\gamma { \ell_1\over 3} }. \leqno (1.46) $$ By the proof of Proposition 1.5 - see Proposition 3.17 in \S3 - there are subintervals $U^- , U^+$ of length 3 in $[\beta_{i-1}, \beta_{i-1} + {\ell_1\over 3} ]$, $[\beta_{i-} - {\ell_1\over 3}, \beta_i ]$ in which $$ |((\eta (s^*) - Q_k ) (t)| \leq 2 {r\over \ell_1^{1/2} } . \leqno (1.47) $$ Hence for $t \in U^\pm$, $$ |\eta (s^*) (t) | \leq 2A\ell_1^{-\gamma {\ell\over 3} } + 2 r\ell_1^{-{1\over 2}} \leqno (1.48) $$ Suppose $U^- = [\alpha^- , \alpha^- + 3]$, $U^+ = [\alpha^+ , \alpha^+ + 3]$. Define $P(t)$ as follows: $$ P(t)=\cases{ 0, & $ t \in (-\infty , \alpha^- + 1] \cup [\alpha^+ + 2,\infty)$ \cr \eta (s^*)(t)& $t \in [\alpha^- + 2, \alpha^+ + 1] $ \cr (t - (\alpha^- + 1)) \eta (s^*) (t)& $t \in (\alpha^- + 1, \alpha^- + 2)$\cr (\alpha^+ + 2 - t) \eta (s^*) (t)& $t\in (\alpha^+ + 1, \alpha^+ + 2)$\cr} \leqno (1.49)$$ Then by the remarks following (1.44), $$ d(P) = d(Q) > 0 \leqno (1.50) $$ so $P \in \Gamma^+$. Moreover for $\ell_1$ sufficiently large, $$ | \int^{\alpha^+ + 2}_{\alpha^- +1} [\LL (P) - \LL (\eta (s^*))]\, dt\mid < {\epsilon\over 2} \leqno (1.51) $$ via (1.48) - (1.49). Hence by (1.49) and (1.51), $$ \leqalignno { I(P) & = \Phi_i (P) < \int^{\alpha^+ +2}_{\alpha^- + 1} \LL (\eta (s^*)) \, dt + {\epsilon\over 2} & (1.52)\cr & < \Phi_i (\eta (s^*)) + {\epsilon\over 2} = c - {\epsilon\over 2} < c\cr} $$ contrary to the definition of $c$. Thus $\chi (\eta (s)) \equiv 1.$ \bigskip \noindent {\bf Remark 1.53.} If $i = 1$ or $k = 1$, the above construction simplifies a bit. It remains to prove that $\psi (\eta (s)) \equiv 1$. Thus suppose that $$ \Phi_i (\eta (s^*)) = c^* + \epsilon . \leqno (1.54) $$ If $\rho \leq \| \eta (s^*) - Q_k \|_{W^{1,2} [\beta_{i-1}, \beta_i]} \leq r$, by (1.9) - (1.10), $$ {d\Phi_i (\eta (s^*))\over ds } = - \Phi'_i (\eta (s^*)) \VV (\eta (s^*)) < 0 \leqno (1.55) $$ But then $\Phi_i (\eta (s))$ is decreasing for $s$ near $s^*$, contrary to the definition of $s^*$. Consequently $$ \| \eta (s^*) - Q_k \|_{W^{1,2} [\beta_{i-1}, \beta_i ]} < \rho . \leqno (1.56) $$ We will show (1.56) is incompatible with (1.54). Define $$ Y(t)=\cases{ \tau_{\sigma_{j_i} } Q (t) & $ t \not\in [\beta_{i-1}, \beta_i] $\cr (\beta_{i-1} + 1-t) \tau_{\sigma_{j_i}} Q(t) + (t - \beta_{i-1} ) \eta (s^*) (t)& $ t \in [\beta_{i-1}, \beta_{i-1} + 1]$\cr \eta (s^*) (t)& $t \in [\beta_{i-1} + 1, \beta_i - 1]$\cr (\beta_i-t ) \eta (s^*) (t) + (t - (\beta_i - 1)) \tau_{\sigma_{j_i}} Q(t) & $t \in [\beta_{i} - 1, \beta_i ]$\cr}\leqno (1.57) $$ Then a computation shows $$ \| Y - \tau_{\sigma_{j_i}} Q\| \leq 3 \| \tau_{\sigma_{j_i}} Q - \eta (s^*) \|_{ W^{1,2} [\beta_{i-1}, \beta_i ] }. \leqno (1.58) $$ Using (1.57) and (1.58) and Proposition 1.11 shows $$ \leqalignno { \| Y & - \tau_{\sigma_{j_i}} Q\| \leq 3\rho + \| Q_k - \tau_{\sigma_{j_i}} Q\|_{W^{1,2} [\beta_{i-1} , \beta_i] } & (1.59) \cr & \leq 3\rho + 3Ae^{-\gamma{\ell_1\over 3} } < 4\rho\cr} $$ for $\ell_1$ sufficiently large. Set $Y \equiv \tau_{\sigma_{j_i} } P$. Therefore by (1.59), $$ \| P - Q\| < 4\rho , \leqno (1.60) $$ i.e. $P \in B_{4\rho} (Q)$ so by (1.28), $$ I(P) \leq c + {\epsilon \over 8} \leqno (1.61) $$ Consequently $$ \leqalignno { I(Y) & = I(P) + \int_\Bbb R (a - \tau_{-\sigma_{j_i}} a)W(P)\, dt & (1.62)\cr &\leq I (P) + {\epsilon\over 2} \leq c + {5\epsilon\over 8} \cr} $$ if $\ell_1$ is sufficiently large. On the other hand, $$ \leqalignno { | \Phi_i & (\eta (s^*)) - I(Y) | \leq \int_{\Bbb R \backslash [\beta_{i-1}, \beta_i]} \LL (Y)\, dt & (1.63) \cr & + \left| \int^{\beta_{i-1} + 1}_{\beta_i - 1} (\LL (\eta (s^*)) - \LL (Y)) \, dt \right| \cr & + \left| \int^{\beta_i}_{\beta_i-1} (\LL (\eta (s^*)) - \LL (Y)) \, dt \right| .\cr} $$ By Proposition 1.11, $$ \int_{\Bbb R \backslash [\beta_{i-1} , \beta_i]} \LL (Y )\, dt \leq A_1 e^{\gamma {\ell_1\over 3} }\leqno (1.64) $$ where $A_1$ depends on $A$ and $\| a\|_{L^\infty}$. Using (1.57) $$ \leqalignno { | \int^{\beta_{i-1} + 1}_{\beta_{i-1}} [\LL (\eta (s^*)) - \LL (Y)]\, dt| \leq &\| \dot \eta \|_{L^2 [\beta_{i-1}, \beta_i]} \| \tau_{\sigma_{j_i} } Q - \eta (s^*) \|_{W^{1,2} [\beta_{i-1}, \beta_i]} \cr & + \| \tau_{\sigma_{j_i}} Q - \eta (s^*) \|^2_{W^{1,2} [\beta_{i-1}, \beta_i ]} &(1.65)\cr & + M_2 \| \tau_{\sigma_{j_i}} Q - \eta (s^*)\|_{W^{1,2} [\beta_{i-1}, \beta_i ] } \cr} $$ where $M_2$ depends on $\| a\|_{L^\infty}$ and $L^\infty$ bounds for $W'$ in a neighborhood of $Q$. Using (1.56) and Proposition 1.11 then gives $$ | \Phi_i (\eta(s^*)) - I(Y) | \leq A_1 e^{-\gamma{\ell_1\over 3}} + A_2 (\rho + A_e^{-{\gamma \ell_1\over 3} } ) + (\rho + (M + M_2) Ae^{-\gamma{\ell_1\over 3} } ) \leqno(1.66) $$ where $A_2$ depends on $\| Q_k \|_{W^{1,2} [\beta_{i-1}, \beta_i] } \leq 2 \| Q\| $. Making $\rho$ possibly still smaller shows $$ | \Phi_i (\eta (s^*)) - \Phi (Y) | \leq {\epsilon \over 4} \leqno (1.67) $$ Consequently by (1.67) and (1.62), $$ \Phi_i (\eta (s^*)) \leq {\epsilon\over 4} + c + {5\epsilon\over 8} = c + {7\over 8} \epsilon < c + \epsilon \leqno (1.68) $$ contrary to (1.54). Thus $\psi (\eta (s)) \equiv 1$ and $I$ must have a local minimizer in $\BB_\rho (Q_k)$. The proof of Theorem 1.2 is complete with $\ell = \ell_1$. \bigskip \centerline{\bf \S2. Related results} This section treats some variants and extensions of Theorem 1.2. In particular, the existence of infinite bump solutions of (HS) will be obtained and the effect of having a pair of isolated minimizers $Q^+, Q^-$ for (0.3) will be studied. To get infinite bump solutions of (HS), let $(\sigma_m)$ be as in (0.4) and let $(\sigma_{j_i} )$ be a subsequence of $(\sigma_m)$ satisfying $\sigma_{j_1} \geq \ell (r)$, $\sigma_{j_{i+1} } - \sigma_{j_i} \geq \ell (r)$ with $r_0, r, \ell (r)$ as given by Theorem 1.2. Let $\beta_i = {1\over 2} (\sigma_{j_i} + \sigma_{j_{i+1}} ), i \in \Bbb N$ and $\beta_0 = - \infty$. Suppose $Q^+ \in \Gamma^+$ is an isolated critical point of $I$ with $I(Q^+) = c^+$. Then for each $k \in \Bbb N$, Theorem 1.2 provides a homoclinic solution $P_k$ of (HS) satisfying $$ \| P_k - \tau_{\sigma_{j_i}} Q^+ \|_{W^{1,2} [\beta_{i-1}, \beta_i]} \leq r, \;\;\; 1 \leq i \leq k - 1 \leqno (2.1) $$ and $$ \| P_k - \sigma_{j_{k-1}} Q^+ \|_{W^{1,2} [\beta_{k-1}, \infty ]} \leq r .\leqno (2.2) $$ By (2.1), the functions $(P_k)$ are bounded in $W^{1,2}_{\rm loc}$ and therefore in $L^\infty_{\rm loc}$. Since they are solutions of (HS), this yields bounds for $(P_k)$ in $C^2_{\loc}$. Hence along a subsequence, $P_k$ converges to a solution, $P$ of (HS) satisfying $$ \| P - \tau_{\sigma_{j_i}} Q^+ \|_{W^{1,2} [\beta_{i-1},\beta_i ]} \leq r \qquad i \in \Bbb N \leqno (2.3) $$ Thus $P$ is an infinite bump solution of (HS) with $|P(t)|, |\dot P(t)| \rightarrow 0$ as $t \rightarrow -\infty$. We state this somewhat informally as \bigskip \proclaim{Theorem 2.4}. Under the hypotheses of Theorem 1.2, for any subsequence $(\sigma_{j_i})$ of $(\sigma_m)$ satisfying $\sigma_{j_1} \geq \ell (r), \sigma_{j_{i+1} } - \sigma_{j_i} \geq \ell(r)$, there is a solution $P$ of (HS) satisfying (2.3). Observe that whenever $Q^-\in \Gamma^-$ is an isolated critical point of $I$ with $I(Q^-) = c^-$, Theorem 1.2 holds with $Q^+$ replaced by $Q^-$. Suppose that both $Q^+$ and $Q^-$ are isolated minimizers of $I$. Then a stronger version of Theorem 1.2 obtains. Indeed let $Y_i \in \{ Q^+, Q^-\}$, $ 1 \leq i \leq k$ and set $X_k = \sum^k_1 \tau_{\sigma_{j_i} } Y_i$. \bigskip \noindent {\bf Theorem 2.5}. Let $(a_1), (W_1) - (W_5)$ be satisfied. Suppose that $I(Q^+) = c^+$, $I(Q^-) = c^-$ with $Q^\pm \in \Gamma^\pm$ and $Q^\pm$ isolated critical points of $I$. Let $k \in \Bbb N$ and $\sigma_{j_i} < \cdots < \sigma_{j_k} \in (\sigma_m)$. Then there is an $r_0 > 0$ and an $\ell = \ell (r)$ defined for $0 < r < r_0$ such that whenever $\sigma_{j_1} \geq \ell$, $\sigma_{j_{i+1} } - \sigma_{j_i} \geq \ell, 1 \leq i \leq k - 1$, and $X_k = \sum^k_1 \tau_{\sigma_{j_i}} Y_i$ with $Y_i \in \{ Q^+, Q^-\}$, $I$ possesses a local minimizer in $\BB_r (X_k)$. \bigskip \noindent {\bf Proof}: The proof requires minor modifications from that of Theorem 1.2 and will be sketched. Suppose $Y_i = Q^+$ for $m$ values of $i$. Then (1.15) becomes $$ I(X_k) \leq m(c^+ + {\epsilon\over 2} ) + (k - m) (c^- + {\epsilon\over 2} )\leqno (2.6) $$ Similarly (1.23) becomes $$ c^\pm - {\epsilon\over 4} \leq \Phi_i (X_k) \leq c^\pm + {\epsilon \over 4} \leqno (2.7) $$ when $Y_i = Q^\pm$ with analogous changes in (1.24) - (1.25). We replace (1.28) by the two conditions $$ \sup_{\overline B_{4_\rho} (Q^\pm)} I(x) \leq c^\pm + {\epsilon\over 8} \leqno (2.8) $$ and $c$ in (1.32) and (1.37) is $c^+$ or $c^-$ depending on $i$. The construction of $P$ following (1.43) is modified to yield a $P^+$ and $P^-$ in $\Gamma^+$ or $\Gamma^-$ with $I(P^\pm ) < c^\pm$. Lastly in (1.54) and the construction of $Y$, $\pm$ cases must be distinguished leading to a contradiction of (2.8). \bigskip \centerline {\bf \S3. Some technical results}\medskip This section contains the proofs of Proposition 1.4, 1.5, 1.11, and 1.13. With the exception of Proposition 1.5, they are quite straightforward so that result will be proved last. \bigskip \noindent{\bf Proof of Proposition 1.4}: Since $Q$ is an isolated critical point of $I$, it can be assumed that $$ \overline B_{r_1} (Q) \cap \KK = \{ Q\} . \leqno (3.1) $$ If Proposition 1.4 is false, there is a sequence $(x_m) \subset \overline B_r (Q) \backslash B_{\underline r} (Q)$ such that $I'(x_m) \rightarrow 0$. For $r_1$ small, $I(x_m)$ is near $c^+$. Therefore, along a subsequence, $I(x_m) \rightarrow b > 0$. Consequently $(x_m)$ is a Palais-Smale sequence. The behavior of such sequences has been studied in [1]. Let $\HH (a)$ denote the closure (in $\| \cdot \|_{L^\infty (\Bbb R )} $) of the set of uniform limits of translates of $a$. For $\alpha \in \HH (a)$, set $$ I_\alpha (x) = \int_\Bbb R \left( {1\over 2} |\dot x|^2 - \alpha W(x)\right)\, dt $$ with associated Hamiltonian system $$ \ddot x + \alpha W' (x) = 0 . $$ Let $$ \KK^* = \{ q \in E\backslash \{ 0 \} | I'_\alpha (q) = 0 \quad \hbox {for some }\quad \alpha \in \HH (a) \} . $$ By Proposition 2.7 of [1], if $(x_m)$ is a Palais-Smale sequence for $I$, there is a\break $j \in \Bbb N, v_1, \cdots , v_j \in \KK^*$, and sequences $(k_m^1) , \cdots (k^j_m) \subset \Bbb R$ such that, along a subsequence, as $m \rightarrow \infty$. $$ \left\| x_m - \sum^j_1 \tau_{k^i_m} v_i \right\| \rightarrow 0\leqno (3.2) $$ and $$ |k^i_m - k^p_m | \rightarrow \infty \quad \hbox {if} \quad i \not = p . \leqno (3.3) $$ Since $$ \| x_m - Q\| \leq r , \leqno (3.4) $$ (3.2) and (3.4) imply $$ \overline {\lim}_{m \rightarrow \infty} \left\| Q - \sum^j_1 \tau_{k^i_m} v_i \right\| \leq r \leq r_1 \leqno (3.5) $$ It was shown in [1 - Remark 2.6] that there is an $r_2 > 0$ so that $\| v\| \geq r_2 $ for all $v \in \KK^*$. Hence for e.g. $r_1 < {1\over 2} r_2$, (3.5) shows $j = 1$ and $(k^1_m )$ is bounded. Therefore by (3.2), $x_m \rightarrow \tau_k v_1 \in \overline B_r (Q) \backslash B_{\underline r} (Q)$. Moreover $I' (\tau_k v_1) = 0$ so $\tau_k v_1 \in \KK$, contrary to (3.1). The Proposition is proved. \bigskip \noindent {\bf Proof of Proposition 1.11}: By $(W_4)$, there are constants $a, \beta > 0 $ such that $|x| \leq \alpha$ implies $$ -x \cdot W'(x ) \geq \beta |x|^2 \leqno (3.6) $$ Set $y (t) = |P(t)|^2$. By (HS) and (3.6), $$ \leqalignno { - \ddot y & = -2 |\dot P |^2 - 2P \cdot \ddot P = -2 |\dot P |^2 + 2aP \cdot W' (P) & (3.7)\cr & \leq -2 |\dot P|^2 - 2a \beta |P|^2.\cr} $$ Define $$ Ly \equiv - \ddot y + 2a_0 \beta y . $$ Then (3.7) and $(a_1)$ show $$ Ly = - | \dot P|^2 + 2(a_0 - a) \beta |P|^2 \leq 0 . \leqno (3.8) $$ Let $\epsilon > 0$, $\gamma_0 = \sqrt {2a_0\beta }$ and $A_0 = \alpha \exp \gamma_0 R$. Define $$ z_\epsilon (t) = A_0 e^{-\gamma_0 t} + \epsilon \leqno (3.9) $$ Then for any $S > R $, $$ L(z_\epsilon - y) = 2a_0 \beta \epsilon - Ly \geq 0, \quad t \in (R,S)\leqno (3.10) $$ and $$ z_\epsilon (R) - y(R) = \alpha + \epsilon - y(R) \geq 0 \leqno (3.11) $$ $$ z_\epsilon (S) - y(S) \geq \epsilon - y(S) \geq 0 \leqno (3.12) $$ for $R$ sufficiently large (since $|P(t)| \rightarrow 0$ as $|t| \rightarrow \infty$). Consequently by the Maximum Principle, $y(t) = |P(t)|^2 \leq z_\epsilon (t)$ for all $t \in [R,S]$. Letting first $S \rightarrow \infty$ and then $\epsilon \rightarrow 0$ shows an estimate of the desired form holds for $|P(t)|$ for $t > R$. Similarly it holds for $t < -R$. By (HS), $$ |\ddot P (t) | \leq a(t) | W'(P) (t)| \leqno (3.13) $$ so $(W_4)$ and the decay estimate for $P$ yield a similar estimate for $\ddot P$. Finally standard interpolation inequalities give the decay estimate for $\dot P$. The proof is complete. \bigskip \noindent {\bf Proof of Proposition 1.13}: Let $(q_m)$ be a minimizing sequence for $I$ in $\overline \BB_\alpha (Q_k)$. Since $(q_m)$ is bounded, it possesses a subsequence converging weakly in $E$ and strongly in $L^\infty_{\loc}$ to $P\in E$. The set $\overline \BB_\alpha (Q_k)$ is closed and convex. Therefore it is weakly closed and $P \in \overline \BB_\alpha (Q_k)$. Moreover for any $\ell > 0$, $$ \int^\ell_{-\ell} \LL (q_m) \, dt \leq I (q_m) \leqno (3.14) $$ so $$ \int^\ell_{-\ell} \LL (P) \, dt \leq \lim_{m\rightarrow \infty} I(q_m) .\leqno (3.15) $$ Letting $\ell \rightarrow \infty$ shows $$ I(P) \leq \lim_{m\rightarrow 0} I(q_m) \leqno (3.16) $$ Consequently $I(P) = \lim_{m\rightarrow\infty} I(q_m)$ and the Proposition is proved. Lastly the proof of Proposition 1.5 will be given. This result is the analogue in the current setting of related results that can be found e.g. in [12] and [7]. The key technical step is its proof is the following: \bigskip \proclaim{Proposition 3.17}. Under the hypotheses of Proposition 1.4, there is an $r_2 > 0$, a function $\ell_1 (r)$ defined for $0 < r \leq r_2$ and a $\varphi_x \in E$ with $\| \varphi_x \| = 1$ defined for $x \in \overline \BB_r (Q_k) \backslash \BB_{r\over 2} (Q_k)$ such that $$ I'(x) \varphi_x \geq 2\delta (6r, {r\over 8} ) \leqno (3.18) $$ ($\delta$ being as in Proposition 1.4) provided that $j_1 \geq \ell_1, j_{i+1} - j_i \geq \ell_1, 1 \leq i \leq k-1 .$ \bigskip \noindent{\bf Proof}: If $x \in \overline \BB_r (Q_k) \backslash \BB_{r/2} (Q_k) ,$ then $x - Q_k \equiv z \in \overline \BB_r (0) \backslash \BB_{r/2} (0) $. Set $z_i = z|_{ [\beta_{i-1}, \beta_i ]}$. Then $$ \| z_i \|_{W^{1,2} [\beta_{i-1}, \beta_i ] } \leq r \qquad 1 \leq i \leq k \leqno (3.19) $$ and for some $p \in [1,k] \cap \Bbb N $, $$ \| z_p\|_{W^{1,2} [\beta_{p-1}, \beta_p ] } \geq {r\over 2} . \leqno (3.20) $$ Assume for convenience that $\ell_1$ is an integer multiple of 12. By (3.19), there is an interval $U^+_i = [s^+_i , s^+_i + 3] \subset [\beta_i - {\ell_1\over 4}, \beta_i ]$ such that $$ \| z_i \|_{W^{1,2} [U^+_i ] } \leq \sqrt {12} r \ell_1^{-1/2} \leqno (3.21) $$ Similarly there is an interval $U^-_i = [s^-_i , s^-_1 + 3] \subset [\beta_{i-1} , \beta_{i-1} + {\ell_1\over 4} ] $ such that $$ \| z_i \|_{W^{1,2} [U^-_i] } \leq \sqrt {12} r\ell_1^{-1/2} . \leqno (3.22) $$ Set $ i = p$ and define a function $z^* (t)$ as follows: $$ z^* (t)=\cases { 0 & $t \in$ center third of $U^+_{p-1}, U^\pm_p, U^-_{p+1}$\cr &\cr 0 & $t \leq s^+_{p-1} + 1 $ and $t \geq s_{p+1} + 2$\cr &\cr z(t) & $t \in [s^+_{p-1} + 3, s^-_p ] \cup [s^-_p + 3, s^+_p] \cup [s^+_p + 3, s^-_{p+1} ]$\cr &\cr (t - (s^+_{p-1} + 2)) z (t) & $t \in [s^+_{p-1} + 2, s^+_{p-1} + 3 ]$\cr &\cr (s^-_p + 1 - t) z(t) & $t \in [s^-_p , s^-_p + 1 ]$\cr &\cr (t - (s^-_p + 2)) z(t)& $t \in [s^-_p + 2, s^-_p + 3 ]$\cr &\cr (s^+_p + 1 - t) z(t)& $t \in [s^+_p , s^+_p + 1]$\cr &\cr (t - (s^+_p + 2)) z(t)& $t \in [s^+_p + 2, s^+_p + 3)$\cr &\cr (s^-_{p+1} + 1 - t) z(t)& $t \in [s^-_{p+1}, s^-_{p+1} + 1 ] $\cr} \leqno (3.23) $$ (If $p = 1$ we need only deal with $U^+_1$ and $U_2^-$ while if $p = k$, $U^+_{k-1}$ and $U^-_k$ suffice). In any of the intervals $U = U^+_{p-1}, U^\pm_p, U^-_{p+1} ,$ $$ \| z - z^* \|_{W^{1,2} [U]} \leq 3 \sqrt {12} r\ell^{-{1\over 2}}_1 . \leqno (3.24) $$ Let $\varphi \in E$ with $\| \varphi \| = 1$ and $\varphi$ having support in $X_p \equiv [s^+_{p-1} , s^-_{p+1} + 3 ]$. Then $$ \leqalignno { I' (Q_k + z)\varphi & = I'(\tau_{\sigma_{j_p}} Q + z^* )\varphi & (3.25)\cr & + (I'(\tau_{\sigma_{j_p}} Q + z) - I' (\tau_{\sigma_{j_p}} Q + z^*))\varphi\cr & + (I' (Q_k + z) - I' (\tau_{\sigma_{j_p}} Q + z))\varphi\cr} $$ Now on $X_p$, $z$ and $z^*$ differ only on $$ \hat U = U^+_{p-1} \cup U^-_p \cup U^+_p \cup U^-_{p+1} . $$ Therefore by (3.24), $$ \leqalignno { |(I'& (\tau_{\sigma_{j_p}} Q + z) - I' (\tau_{\sigma_{j_p} } Q + z^*)) \varphi | & (3.26) \cr & = | \int_{\hat U} [ (\dot z - \dot z^*) \cdot \dot\varphi - a(t) (W' (\tau_{\sigma_{j_p}} Q + z) - W' (\tau_{\sigma_{j_p}} Q + z^* )) \cdot \varphi ] \, dt\cr & \leq M_1 \| z - z^* \|_{W^{1,2} [\hat U]} \leq 12 \sqrt {12} r\ell_1^{-{1\over 2}} M_1 \equiv M_2 r\ell_1^{-{1\over 2}}\cr} $$ where $M_1$ depends on $\| a\|_{L^\infty}$ and $L^\infty$ bounds for the second derivatives of $W$ in a $(2r_1)$ neighborhood of $Q$. Hence for $\ell_1 (r)$ sufficiently large. $$ | (I' (\tau_{\sigma_{j_p}} Q + z) - I' ( \tau_{\sigma_{j_p}} Q + z^*)) \varphi | \leq {1\over 4} \delta (6r, {r\over 8}).\leqno (3.27) $$ The next difference on the right in (3.25) can be estimated as follows: $$ \leqalignno { |(I' (Q_k &+ z) - I' ( \tau_{\sigma_{j_p}} Q + z)) \varphi |& (3.28) \cr & = | \int_{X_p} [ \sum_{i\not = p} \tau_{\sigma_{j_i}} \dot Q \cdot \dot \varphi - a(t) (W' (Q_k + z) - W' (\tau_{\sigma_{j_p}}Q+z)) \cdot \varphi \, dt \cr & \leq (1 + M_1) \sum_{i\not = p} \| \tau_{\sigma_{j_i}} Q\|_{W^{1,2} (X_p)} \cr} $$ where $M_1$ is as above. It can be assumed that in Proposition 1.11, $R < \ell_1 /4$. Therefore $t \in X_p$ and $i\not = p$, so by Proposition 1.11, $$ | \tau_{\sigma_{j_i}} \dot Q (t)|, | \tau_{\sigma_{j_i}} Q(t)| \leq A_e^{ - \gamma |t - \sigma_{j_i} |}. \leqno (3.29) $$ Hence the decay estimate together with the choice of the $\sigma_{j_i}$'s: $\sigma_{j_{i+1}} - \sigma_{j_i} \geq \ell_1$ yields $$ \leqalignno { \sum_{i \not = p} & \| \tau_{\sigma_{j_i}} Q\|_{W^{1,2} (X_p)} \leq {A\over \gamma^{1/2}} \sum^k_1 e^{-{\gamma \ell_1 i \over 4}} & (3.30)\cr & \leq {2Ae^{-{\gamma \ell_1 \over 4} } \over \gamma^{1/2} (1 - e^{-{\gamma\ell_1\over 4}} ) } \leq {1\over 4} \delta (6r, {r\over 8} )\cr} $$ for $\ell_1 $ sufficiently large. Combining (3.27) and (3.30) gives $$ I'(Q_k + z) \varphi \geq I' ( \tau_{\sigma_{j_p}} Q + z^*) \varphi - {1\over 2} \delta (6r, {r\over 8} ) \leqno (3.31) $$ Now (3.18) can be obtained by making an appropriate choice of $\varphi$ in (3.31). Since $z$ and $z^*$ differ on $X_p$ only on the region $\hat U$ where the difference is small, (3.24), and (3.19) - (3.20), show for $\ell_1$ sufficiently large. $$ 2r \geq \|z^*_p \|_{W^{1,2} [\beta_{p-1}, \beta_p] } \geq {r\over 4} . \leqno (3.32) $$ Set $Y_p = [s^-_p + 1, s^+_p + 2] \subset [\beta_{p-1}, \beta_p ] $. Two cases will be considered. Suppose that $$ \| z^*_p \|_{W^{1,2} (Y_p)} \geq {r\over 8} .\leqno (3.33) $$ Define $$ Z_p (t) = \cases{z^*_p (t)& $t \in Y_p$ \cr 0 & $t \in R\backslash Y_p $\cr}\leqno (3.34) $$ Then $Z_p \in E$ and by construction, $Z_p \in \overline B_{6r} (0) \backslash B_{r\over 8} (0)$. Therefore by Proposition 1.4, $$ \| I' (\tau_{\sigma_{j_p}}Q+Z_p) \| \geq 4\delta (6 r, {r\over 8}) \leqno (3.35) $$ for $r_2$ appropriately small. Hence there is a $\varphi \in E$ with $\| \varphi \| = 1$ such that $$ I'( \tau_{\sigma_{j_p}} Q + Z_p) \varphi \geq 3\delta (6r, {r\over 8} ) .\leqno (3.36) $$ Moreover since the support of $Z_p$ lies in $Y_p$ and $\tau_{\sigma_{j_p}} Q$ decays exponentally outside of an ${\ell_1\over 4}$ neighborhood of $\sigma_{j_{p}}$ via Proposition 1.11, it can be assumed that $\varphi$ has support in $Y_p$. Therefore since $z^* = z^*_p$ on $Y_p$, $$ I' ( \tau_{\sigma_{j_p}} Q + Z_p) \varphi = I' ( \tau_{\sigma_{j_p}} Q + z^* ) \varphi \geq 3\delta (6r, {r\over 8}) \leqno (3.37) $$ and (3.18) obtains for this case with $\varphi_x= \varphi$. \bigskip \noindent {\bf Remark 3.38.} For future reference, observe that the above arguments also yield $$ \Phi'_p (x) \varphi_x \geq 2\delta (6r , {r\over 8} ) \leqno (3.39) $$ for this case. Next suppose that $$ \| z^*_p \|_{W^{1,2} (Y_p)} < {r\over 8} . \leqno (3.40) $$ Then by (3.32) - (3.34), $$ \| z^*_p - Z_p \|_{W^{1,2} [\beta_{p-1}, \beta_p ] } > {r\over 8} \leqno (3.41) $$ Set $$ \varphi = (z^* - Z_p) \| z^* - Z_p \|^{-1} \equiv (z^* - Z_p) b . $$ Then $\varphi$ has support in $X_p \backslash Y_p$ and $$ \leqalignno { I' & ( \tau_{\sigma_{j_p}} Q + z^* ) \varphi & (3.42) \cr & = b\int_{X_p \backslash Y_p} ( \tau_{\sigma_{j_p}} \dot Q \cdot \dot z^* + |\dot z^* |^2 - aW' ( \tau_{\sigma_{j_p}} Q + z^*) \cdot z^* ) \, dt\cr & = b \int_{X_p \backslash Y_p } [ |\dot z^* |^2 - a W' (z^* ) \cdot z^*\cr & \qquad + \tau_{\sigma_{j_p}} \dot Q \cdot \dot z^* - a (W' ( \tau_{\sigma_{j_p}} Q + z^* ) - W' (z^* )) \cdot z^* ]\, dt \cr} $$ In the region $X_p \backslash Y_p$, $\tau_{\sigma_{j_p}}Q$ is exponentially small. This yields the estimate $$ \leqalignno { | \int_{X_p \backslash Y_p } & [ \tau_{\sigma_{j_p}} \dot Q \cdot \dot z^* - a (W' ( \tau_{\sigma_{j_p}} Q + z^* ) - W' (z^*)) \dot z^* ] \, dt | & (3.43)\cr & \leq M_3 e^{-\gamma \ell_1/ 4} \| z^* \|_{W^{1,2} [X_p \backslash Y_p] } \cr} $$ where $M_3$ depends on $A, \gamma, \| a\|_{L^\infty }$, and $L^\infty$ bounds for the second derivatives of $W$ in a neighborhood of $0$. Thus by (3.42) - (3.43), $$ \leqalignno { I' ( \tau_{\sigma_{j_p}} Q + z^*) \varphi & \geq b ( \min (a_0, 1)) \|z^* \|_{W^{1,2} [X_p \backslash Y_p ] } & (3.44)\cr & ( \| z^* \|_{W^{1,2} [ X_p \backslash Y_p ]} - M_3 e^{-\gamma\ell_1/4} ) .\cr} $$ Since $$ \| z^* \|_{W^{1,2} [X_p \backslash Y_p ] } \geq \| z^* \|_{W^{1,2} [\beta_{p-1}, \beta_p ] } = \| z^*_p \|_{W^{1,2} [\beta_{p-1}, \beta_p }] \geq {r\over 8} \leqno (3.45) $$ via (3.41), for $\ell_1$ sufficiently large, $$ I' ( \tau_{\sigma_{j_p}} Q + z^* ) \varphi \geq {b\over 16} \min (a_0, 1)r^2 . \leqno (3.46) $$ Finally since $b \geq {1\over 6r},$ $$ I' ( \tau_{\sigma_{j_p}} Q + z^* ) \varphi \geq {1\over 96} \min (a_0, 1)r . \leqno (3.47) $$ It can be assumed that the right hand side of (3.47) is large compared to $\delta (6r, {4\over 8})$. Hence we obtain (3.18) for this case. \bigskip \noindent {\bf Remark 3.48.} Note that for this case by above estimates and the current choice of $\varphi$, $$ \Phi'_p (x) \varphi \geq \Phi'_p ( \tau_{\sigma_{j_p}} Q + z) \varphi - {1\over 2} \delta (6r, {r\over 8} ) . \leqno (3.49) $$ Arguing as in (3.42) - (3.44) gives $$ \leqalignno { \Phi'_p & ( \tau_{\sigma_{j_p}} Q + z) \varphi \geq b \min (a_0, 1) \| z^*_p \|_{W^{1,2} [[\beta_{p-1}, \beta_p]\backslash Y_p] } & (3.50)\cr & \cdot (\| z^*_p \|_{W^{1,2} [[\beta_{p-1}, \beta_p]\backslash Y_p] } - M_3 e^{-\gamma \ell_1 /4} )\cr} $$ so by (3.45) $$ \Phi'_p (\tau_{\sigma_{j_p}} Q + z) \varphi \geq {1\over 16} \min (a_0, 1) r \leqno (3.51) $$ as in (3.47). thus (3.18) obtains for this case also. The proof of Proposition 3.17 is complete. \bigskip \noindent {\bf Remark 3.52.} Having obtained (3.18) for $x \in \overline \BB_r (Q_k) \backslash \BB_{r\over 2} (Q_k)$, replacing $r$ by ${r\over 2^m} , m = 1,2,\cdots ,m_0$ where ${r\over 2^{m_0-1} } > \rho \geq {r\over 2^{m_0}}$ and appropriately adjusting $\ell_1$ yields a $\varphi_x$ for which $$ I' (x) \varphi_x \geq 2\delta (6r, {\rho\over 8} ) \leqno (3.53) $$ for all $x \in \overline \BB_r (Q_k) \backslash \BB_\rho (Q_k) ).$ \bigskip \noindent {\bf Remark 3.54.} In Case 1 of the proof of Proposition 3.17 $$ \| \varphi_x \| = \tv \varphi_x \tv = 1 $$ since the support of $\varphi_x$ lies in $[\beta_{p-1}, \beta_p]$ while in Case 2, the support of $\varphi_x$ may extend into the 2 adjacent intervals. Hence $\tv \varphi_x \tv \leq 1$. If there are several values of $p$ for which (3.20) holds, say $p_1, \cdots , p_n$, take $\varphi_x = \varphi_{x_{p_1}} + \cdots + \varphi_{x_{p_n}}$. Then $\tv \varphi_x \tv \leq 3$ due to our observation about the supports of the functions $\varphi_{x_{p_i}}$. Now finally we have \bigskip \noindent {\bf Proof of Proposition 1.5}: A standard construction using convex combinations of the $\varphi_x$'s and cut-off functions yields $\VV (x)$. See e.g. Lemma A.2 of [22] for details. In particular Remark 3.54 gives (1.6), (3.18) and Remark 3.52 prove (1.7), and Remarks 3.38 and 3.48 give (1.9) - (1.10). \bigskip \centerline {\bf References } \medskip \item{[1]} Rabinowitz, P.\ H., Homoclinics for an almost periodically forced Hamiltonian system, {\it Top. Meth. in Nonlinear Analysis} (to appear). \smallskip \item{[2]} S\'er\'e, E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, {\it Math Z.}, {\bf 209}(1992), 27--42. \smallskip \item{[3]} Bessi, U., A variational proof of a Sitnikov-like Theorem, {\it Nonlinear Analysis}, TMA, {\bf 20}(1993), 1303--1318. \smallskip \item{[4]} Bolotin, S.\ V., {Homoclinic orbits to invariant tori of symplectic diffeomorphisms and Hamiltonian systems}, preprint. \smallskip \item{[5]} Caldiroli, P.\ and P.\ Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign, {\it Comm.\ on App.\ Nonlinear Anal.}, {\bf 1}(1994), 97--129. \smallskip \item{[6]} Coti Zelati, V.\ and P.\ H.\ Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, {\it J.\ Amer.\ Math.\ Soc.},\hfil\break {\bf 4}(1991), 693--727. \smallskip \item{[7]} Coti Zelati, V.\ and P.\ H.\ Rabinowitz, Multibump periodic solutions of a family of Hamiltonian systems, {\it Topological Methods in Nonlinear Analysis}, {\bf 4}(1995), 31--57. \smallskip \item {[8]} Giannoni, F.\ and P.\ H.\ Rabinowitz, On the multiplicity of homoclinic orbits on Riemannian manifolds for a class of Hamiltonian systems, {\it Nonlin. Diff. Eq. and Applic.}, {\bf 1}(1994), 1--46. \smallskip \item{[9]} Montecchiari, P.\ and N.\ Nolasco, Multibump solutions for perturbations of periodic second order Hamiltonian systems, {\it Nonlin.\ Anal.}, TMA (to appear). \smallskip \item {[10]} Rabinowitz, P.\ H., Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, {\it Calc. Var.}, {\bf 1}(1993), 1--36. \smallskip \item{[11]} Rabinowitz, P.\ H., Multibump solutions of differential equations: An overview, {\it Chinese Journal of Mathematics}, to appear. \smallskip \item{[12]} S\'er\'e, E., Looking for the Bernoulli shift, {\it Ann.\ Inst.\ H.\ Poincar\'e, Ann.\ Non Lin\'eaire}, {\bf 10}(1992), 561--590. \item{[13]} Strobel, K.\ H., {Multibump solutions for a class of periodic Hamiltonian}, University of Wisconsin, Thesis, April 1994. \smallskip \item{[14]} Buffoni, B.\ and E.\ S\'er\'e, {A global condition for quasi-random behaviour in a class of conservative systems}, preprint (1995). \smallskip \item{[15]} Rabinowitz, P.\ H., Homoclinics for a singular Hamiltonian system, to appear in {\it Geometric Analysis and the Calculus of Variations}, J.\ Jost (ed), International Press. \smallskip \item{[16]} Tanaka, K., Homoclinic orbits for a singular second order Hamiltonian system, {\it Ann.\ Inst.\ H.\ Poincar\'e, Ann.\ Non Lin\'eaire}, {\bf 7}(1990), 427--438. \smallskip \item{[17]} Serra, E, M.\ Tarallo and S.\ Terracini, {On the existence of homoclinic solutions for almost periodic second order systems}, preprint (1994). \smallskip \item{[18]} Bertotti, M.\ L.\ and S.\ Bolotin, {Homoclinic solutions of quasiperiodic Lagrangian systems}, preprint (1994). \smallskip \item {[19]} Spradlin, G., Private communication. \smallskip \item {[20]} Coti Zelati, V., P.\ Montecchiari, and M.\ Nolasco, {Multibump solutions for a class of almost periodic Hamiltonian systems}, preprint. \smallskip \item{[21]} Caldiroli, P.\ and M.\ Nolasco, {Multiple homoclinic solutions for a class of autonomous singular systems in $\Bbb R^2$}, preprint. \smallskip \item{[22]} Rabinowitz, P.\ H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, C.B.M.S.\ Regional Conf.\ Ser.\ in Math \#65, A.M.S.\ (1986). \bigskip \vbox{Paul H. Rabinowitz \hfil\break Department of Mathematics \hfil\break University of Wisconsin \hfil\break Madison, Wisconsin 53706 \hfil\break E-mail address: rabinowi@math.wisc.edu} \bye