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\markboth{ Multichain-type solutions for Hamiltonian systems }
{ Paul H. Rabinowitz \& Vittorio Coti Zelati}
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Nonlinear Differential Equations, \newline
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 223--235\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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%
 Multichain-type solutions for \\ Hamiltonian systems
% 
\thanks{ {\em Mathematics Subject Classifications:} 34C37, 37J45, 37J50,
58E99.  
 \hfil\break\indent 
{\em Key words:}  multichain solutions, Hamiltonian systems, minimization
methods. 
 \hfil\break\indent
\copyright 2000 Southwest Texas State University. 
\hfil\break\indent Published October 25, 2000.  \hfil\break\indent
(V.C.Z.) was supported by MURST, Programma di richerca scientifica di 
interesse nazionale \hfil\break\indent
``Methode Variazionali e Equazioni Differenziali Nonlineari''
\hfill\break\indent
(P.H.R.) was supported by the NSF under grant MCS8110556 } } 

\date{}
\author{ Paul H. Rabinowitz \& Vittorio Coti Zelati \\ [12pt]
{\em Dedicated to Alan Lazer} \\ {\em on  his 60th birthday }}
\maketitle

\begin{abstract} 
The existence of basic and more complicated multichain heteroclinic
solutions is established for a class of forced slowly oscillating
Hamiltonian systems.  Constrained minimization arguments are the key tool
in obtaining the results.
 \end{abstract}

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\newtheorem{corollary}[theorem]{Corollary} 
\newtheorem{remark}[theorem]{Remark}             

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\section{Introduction}

Consider the Hamiltonian system 
\begin{equation} \label{HS}
\ddot q+V_q(t,q)=0\,,  
\end{equation}
 where $q=(q_1,\ldots ,q_n)\in{\mathbb R}^n$ and $V$ satisfies
 \begin{description}
\item{(V$_1$)} $ V\in C^2 ({\mathbb R}\times {\mathbb R}^n, {\mathbb R})$, 
is $1$-periodic in $t$ and $1$-periodic in $q_i$, $1\le i\le n$;

\item{(V$_2$)} $V(t,0)=0>V(t,x)$ with $x\in{\mathbb R}^n\backslash {\mathbb Z}^n$. 
\end{description}
 This system was studied by Strobel \cite{s1} who proved the  following:
\begin{description}
\item{(a)} For each
$\xi\in{\mathbb Z}^n$, there is an $\eta\in{\mathbb Z}^n\backslash\{\xi\}$ and a solution
$Q$ of (\ref{HS}) heteroclinic from $\xi$ to $\eta$, i.e. $Q(-\infty )=\xi$ and
$Q(\infty )=\eta$ 
\item{(b)} For each $\xi\not=\eta\in{\mathbb Z}^n$, there is a
heteroclinic chain of solutions of (\ref{HS}) joining $\xi$ and $\eta$, i.e.\
there exist $\xi_0=\xi$, $\xi_1,\ldots ,\xi_k=\eta$ and solutions $Q_i$ of
(\ref{HS}) heteroclinic from $\xi_{i-1}$ to $\xi_i$, $1\le i\le k$. 
\end{description}
Earlier versions of (a) and (b) when $V=V(x)$ were obtained in \cite{r1,r3,b4}.  
More recently,
Bertotti and Montecchiari \cite{b3} have treated (\ref{HS}) where $V(t,x)=a(t)W(x)$ with $W$
satisfying (V$_1$)--(V$_2$) and $a$ almost periodic in $t$.  They
also find infinitely many heteroclinic solutions of (\ref{HS}) but without
a 
nondegeneracy condition as in \cite{s1}, they cannot make as precise
existence 
statements as \cite{s1}.

In his setting, under a further nondegeneracy condition involving the
functions $Q_i$ in
(b), Strobel proved that in fact there exist infinitely many solutions of
(\ref{HS}) heteroclinic from $\xi$ to $\eta$ which are near the chain
$Q_1,\ldots ,Q_k$ and are distinguished by the amount of time they spend
near $Q_1(\infty ), \ldots ,Q_{k-1}(\infty )$.

In this paper, results related to \cite{s1} will be proved for two classes of
potentials that are of a more restricted form than $V(t,x)$, namely
$a(t)W(x)$.  However $a(t)$ is not necessarily periodic in $t$ and unlike
\cite{s1}, no nondegeneracy conditions will be required.  The function $W$
satisfies the time independent version of (V$_1$)--(V$_2$):
\begin{description}
\item{(W$_1$)} $W\in C^2({\mathbb R}^n, {\mathbb R})$ is $1$-periodic in $q_i$, 
and $1\le i\le n$;

\item{(W$_2$)} $W(0)=0>W(x)$ with $x\in{\mathbb R}^n\backslash{\mathbb Z}^n$.
\end{description}

\noindent
For the first class of potentials, roughly speaking,  $a(t)$ is nearly
constant near a sequence of its local maxima and minima which are
sufficiently far apart.  This will be made precise in \S2.  For example if
$a(t)$ is $1$-periodic, continuous, positive, and non-constant, for all
small $\epsilon>0$, $a(\epsilon t)W(x)$ will be an allowable potential.  A second
class of potentials are of the form $(\alpha_1(\epsilon t)+\alpha_2(t))W(x)$
where $\alpha_1, \alpha_2$ are e.g.\ each like the $a$ just described.

Bolotin and MacKay \cite{b6} have recently studied multichain type solutions for
a class of slowly oscillating problems in a setting that is more general
than ours  in some
ways but less general in particular in $t$ dependence.  Their approach involves
a mixture of analytical and minimization arguments.
In very recent work,  Alessio, Bertotti, and Montecchiari
\cite{a4}
studied a generalization of \cite{b3} and also showed that by perturbing such a
situation by a term of the form $\alpha (\epsilon t)W(x)$ with $\alpha$
almost
periodic and $\epsilon$ small, they get solutions of multichain type.  Although
there is some intersection with this paper, the point of view taken here is
quite different from that of \cite{a4}.  For other related results in a small
perturbation setting, see Ambrosetti and Badiale \cite{a1}, Ambrosetti and Berti
\cite{a3}, Berti \cite{b1}, Berti and Bolle \cite{b2}.

It is also worth noting that there has been a considerable amount of work in a
PDE setting on standing wave solutions for nonlinear Schr\"odinger equations
which have slowly oscillating spacially dependent potentials.  See e.g.\ Floer
and Weinstein \cite{f1}, Oh \cite{o1}, Thandi \cite{t1}, del~Pino and Felmer 
\cite{d1},  and Ambrosetti, Badiale, and Cingolani \cite{a2}
to mention a
few.

In \S2, the existence of basic heteroclinic solutions will be established.
The existence of heteroclinics near finite chains of basic solutions will
be given in \S3.  Some simple observations then yield the case of
solutions of infinite chain type.  The proofs involve elementary
minimization and comparison arguments.


\section{Basic heteroclinic solutions}


In this section, the existence of basic heteroclinic orbits will 
be established.  To begin, let $$ {\mathcal A} =\{ a\in C({\mathbb R}, 
{\mathbb R})\mid 0<\underline a\le a(t)\le \overline a<\infty\} $$ where 
$\underline a <\overline a$.  Our first goal is to find a solution 
of (\ref{HS}) heteroclinic from $0$ to some $\xi\in{\mathbb 
R}^n\backslash\{ 0\}$. Choose $r>0$ which is small compared to 
$1\equiv \inf\{ |\xi_i-\xi_j|\mid \xi_i\not= \xi_j\in{\mathbb Z}^n\}$, 
i.e.\ $r\ll 1$.  A further condition will be imposed on $r$ later.  
Let $B_r(z)$ denote an open ball of radius $r$ about $z\in{\mathbb 
R}^n$. Let $b_1<b_2-1$.  A heteroclinic solution of (\ref{HS}) will be 
obtained such that the transition between the end states occurs 
mainly in $[b_1,b_2]$. Define 
\begin{eqnarray*} 
\Gamma =\Gamma (b_1,b_2)&=&\big\{ q\in W^{1,2}_{\rm loc} ({\mathbb R},{\mathbb R}^n)
: q(t)\in\overline B_r(0), t\le b_1,\\
&&\mbox{ and }q(t)\in\overline B_r(\xi )\mbox{ for some
$\xi\in{\mathbb Z}^n\backslash\{ 0\}$, $t\ge b_2$}\big\}.
\end{eqnarray*}
Set
$$
L(q)={1\over 2} |\dot q|^2-a(t)W(q),
$$
the Lagrangian for (\ref{HS}), and define the associated functional
$$
I(q)=\int_{\mathbb R} L(q)dt.
$$
Finally define
\begin{equation} \label{2.1}
c=c(b_1, b_2)=\inf_{q\in\Gamma} I(q)\,.
\end{equation}


\begin{proposition} \label{p2.2} 
If $a\in {\mathcal A}$ and $W$ satisfies (W$_1$)--(W$_2$), there 
is a $Q\in \Gamma (b_1,b_2)$ such that $I(q)=c(b_1,b_2)$.
\end{proposition}

\paragraph{Proof:}  Let $(q_m)$ be a minimizing sequence for (\ref{2.1}).  Then the
form of $I$ and $\Gamma$ imply $(q_m)$ is bounded in $W^{1,2}_{\rm loc}$
and converges weakly in $W^{1,2}_{\rm loc}$ and strongly in $L^\infty_{\rm
loc}$ to $Q\in\Gamma (b_1,b_2)$.  Moreover standard weak lower
semicontinuity arguments imply $I(Q)=c(b_1,b_2)$.


\begin{remark} \rm \label{r2.3}
\begin{description}
\item{(i)} As in \cite{r1}, $Q(-\infty )=0$ and $Q(\infty )\in
{\mathbb Z}^n\backslash \{ 0\}$.


\item{(ii)} Standard regularity arguments show $Q$ is a solution of (\ref{HS})
for $t\in (b_1, b_2)$ and also for those values of $t\le b_1$, $t\ge b_2$
when $Q(t)\not\in \partial B_r(0)$, $Q(t)\not\in\partial B_r(Q(\infty ))$
respectively.
\end{description}
\end{remark}

\noindent It remains to choose a subfamily ${\mathcal 
A}^*\subset{\mathcal A}$ for which $a\in{\mathcal A}^*$ implies 
$Q$ is a solution of (\ref{HS}).  First a few observations about $Q$ are 
necessary.  Suppose $Q(\infty )=\xi$. 


\begin{lemma} \label{l2.4}
For $0<\rho <r$, there is an $\omega =\omega(\rho )>0$ and 
$t_1=t_1(\rho )\in [b_1-\omega ,b_1]$ such that $Q(t_1)\in
\overline B_\rho (0)$.  Moreover $\omega$  can be chosen
independently of $a\in{\mathcal A}$.
\end{lemma}

\paragraph{Proof:} Since $Q(-\infty )=0$, $Q(t)\in B_\rho (0)$ for $t$ near
$-\infty$.  The point is to find $\omega$  independently of
$a\in{\mathcal A}$.  Let $\eta\in{\mathbb Z}^n\backslash \{ 0\}$ and define
\begin{equation} \label{2.5}
R(t)=\cases{0 & if $t\le b_1$, \cr
  (t-b_1)\eta & if $b_1\le t\le b_1+1$,\cr
  \eta        & if $t\ge b_1+1$.}
\end{equation}
Then $R$ belongs to $\Gamma$, so
\begin{equation} \label{2.6}
c=I(Q)\le I(R)\equiv M\,.
\end{equation}
Set
\begin{equation} \label{2.7}
\beta (\rho )=\inf_{|x-{\mathbb Z}^n|\ge\rho} -W(x)\,.
\end{equation}
By (W$_1$)--(W$_2$), $\beta (\rho )>0$.  If $|Q(t)|>\rho$ in $[b_1-
\omega , b_1]$, by (\ref{2.6})--(\ref{2.7}),
$$
M\ge I(\varphi )\ge\int^{b_1}_{b_1-\omega} -a(t)W(Q)dt\ge\underline a\beta
(\rho )\omega\,.
$$
Thus the Lemma holds for any $\omega >M(\underline a\beta (\rho ))^{-1}$.


\begin{corollary} \label{c2.9}
There is a $t_2=t_2(\rho )\in [b_2,
b_2+\omega ]$ such that $Q(b_2)\in\overline B_\rho (\xi )$.
\end{corollary}

\paragraph{Proof:}  As in Lemma \ref{l2.4}. After obtaining $t_1$, we define
\begin{equation} \label{2.10}
P(t)=\cases{ 0    & if $t\le t_1-1$, \cr
(t-(t_1-1))Q(t_1) & if $t_1-1\le t\le t_1$,\cr
Q(t)              & if $t\ge t_1$.}
\end{equation}
Then $P\in\Gamma (b_1, b_2)$ so $I(Q)\le I(P)$ and in particular  by
(\ref{2.10}),
\begin{equation} \label{2.11}
\int^{t_1}_{-\infty} L(Q)dt\le\int^{t_1}_{-\infty}
L(P)dt=\int^{t_1}_{t_1-1} L(P)dt\equiv\varphi (\rho )
\end{equation}
and the definition of $\varphi (\rho )$ shows $\varphi (\rho )\to 0$ as
$\rho\to 0$.  Similarly, it can be assumed that
$$
\int^\infty_{t_2} L(Q)dt\le\varphi (\rho ).
$$


\begin{lemma} \label{l2.13}
For  $\rho\ll r$, $Q(t)\in B_{r/2}(0)$ for $t\le t_1$ and 
$Q(t)\in B_{r/2}(\xi )$ for $t\ge t_2$.
\end{lemma}

\paragraph{Proof:}  The first assertion will be proved.  If it is not valid,
$Q(s)\in\partial B_{r/2}(0)$ for some $s<t_1$.  By Lemma \ref{l2.4},
$Q(t_1)\in\overline B_\rho (0)$.  For $\rho \ll r$, the cost of $Q$ going
from $\partial B_{r/2}(0)$ to $\partial B_\rho (0)$, as measured by
$I$, exceeds that of going from $0$ to $\partial B_\rho (0)$ as in
(\ref{2.10})--(\ref{2.11}).  Since $Q$ minimizes $I$ in $\Gamma$, the Lemma follows.


\begin{lemma} \label{l2.14}
There is an $s_1\in [b_1, b_1+\omega ]$ and
$s_2\in [b_2-\omega, b_2]$ such that $Q(s_1)$, $Q(s_2)\in \overline B_\rho
(0)\cup \overline B_\rho (\xi )$.
\end{lemma}

\paragraph{Proof:}  The proof of Lemma \ref{l2.4} shows there exists $s_i$ with
$Q(s_i)\in\overline B_\rho (x_i)$ for some $x_i\in{\mathbb Z}^n$, $i=1,2$.  Thus
the possibility that $x_i\not\in \{ 0,\xi \}$ must be excluded.  Since
$Q(-\infty )=0$, $Q(b_2)\in\overline B_r(\xi )$, $r\ll 1$, and $Q$ minimizes
$I$ in $\Gamma_1$, simple comparison arguments in the spirit of Lemma \ref{l2.13}
show $x_i=0$ or $\xi$, $i=1,2$.

To show that $Q$ is a solution of (\ref{HS}), further conditions will 
have to be imposed on $a\in{\mathcal A}$: 
 \begin{description} 
\item{(a$_1$)} there is a $T>0$ and a sequence of points
$(m_i)_{i\in{\mathbb Z}}\subset{\mathbb R}$ such that $m_{i+1}-m_i\ge T$

\item{(a$_2$)} there is a  $\gamma >0$ and $\theta_i\in (2\omega ,
m_i-m_{i-1}-2\omega )$, such that for all $i\in{\mathbb Z}$,
where \begin{description}
\item{(i)} $a(t)-a(s)\ge\gamma$, $t\in [m_i-\omega, m_i+\omega ]$,
$s\in [m_i-\theta_i-\omega, m_i-\theta_i+\omega ]$.

\item{(ii)} $a(t)-a(s)\ge\gamma$, $  t\in [m_i-\omega, m_i+\omega ]$,
  $s\in [m_i+\theta_{i+1}-\omega, m_i+\theta_{i+1} +\omega]$.
\end{description}
\end{description}
Define $$ {\mathcal A}^*=\{ a\in{\mathcal A} : \mbox{(a$_1$)  and 
(a$_2$) hold}\}\,. $$ 
 Conditions (a$_1$)--(a$_2$) are satisfied if e.g. $a$ is $T$
periodic in $t$, with $T$ appropriately large, $m_{i+1}=m_i+T$,
$a(m_i)=\max a$, $\theta_{i+1}=\theta_i+T$, $a(m_i+\theta_i)=\min a$,
$\gamma ={1\over 2} (a(m_i)-a(m_i+\theta_i))$ and $a$ oscillates slowly so
(a$_2$) holds.  More generally, it suffices that $a$ remains near its
maximum and minimum on a large time interval.  In particular, as mentioned
in the Introduction, these conditions will be satisfied if $a(t)=b(\epsilon t)$
with $b$ positive, continuous, $1$-periodic in $t$, and $\not\equiv$
constant, and $\epsilon$ sufficiently small.  Suppose further
\begin{equation} \label{2.15}
\varphi (\rho )<{\gamma\over 32M} \big({\underline a/\overline
a}\big) \beta (r)\,.
\end{equation}
Choosing $(b_1, b_2)=(m_i, m_{i+1})$, we have


\begin{theorem} \label{t2.16}
Suppose  (W$_1$)--(W$_2$) hold, $\rho$ and $r$ satisfy $\rho \ll 
r\ll 1$ and (\ref{2.15}), and $a\in{\mathcal A}^*$.  Then for each 
$i\in{\mathbb Z}$, (\ref{HS}) has a solution $Q=Q_i\in\Gamma (m_i, m_{i+1})$ with 
$I(Q_i)=c(m_i, m_{i+1})$. 
\end{theorem}

\paragraph{Proof:} Since it does not effect the argument, for notational
simplicity, we set $i=1$.  By Remark \ref{r2.3} and Lemma \ref{l2.13}, $Q$ is a solution
of (\ref{HS}) except possibly for $t\in (t_1, m_1]\cup [m_2, t_2)$.  Suppose
e.g. $Q(t)\in\partial B_r(0)$ for some $t\in (t_1, m_1]$.  Then the cost
 analysis of
Lemma \ref{l2.13} shows, $Q(s_1)\in\overline B_\rho (\xi )$, $Q(t)\in B_{r\over
2}(\xi )$ for $t\ge s_1$, and
$$
\int^\infty_{s_1} L(Q)dt\le \varphi (\rho )\,.
$$
Therefore, $Q^*(t)=Q(t-\tau )\in\Gamma$ for any $\tau\in [0,m_2-s_1]$.
Since $\theta_2<m_2-m_1-2\omega$ and $s_1<m_1+\omega$,
$\theta_2<m_2-s_1-\omega <m_2-s_1$ so taking $\tau =\theta_2$ shows
\begin{equation} \label{2.18}
0\ge I(Q)-I(Q^*)=-\int_{\mathbb R} (a(t)-a(t+\theta_2))W(Q)\,dt\,.
\end{equation}
Now
\begin{equation} \label{2.19}
\Big|\int^{t_1}_{-\infty} (a(t)-a(t+\theta_2))W(Q)dt\Big|\le 2{\overline
a\over\underline a}\int^{t_1}_{-\infty}L(Q)dt\le 2{\overline
a\over\underline a}\varphi (\rho)
\end{equation}
and similarly
\begin{equation} \label{2.20}
\Big|\int^\infty_{s_1} (a(t)-a(t+\theta_2))W(Q)dt\Big|\le 2{\overline
a\over\underline a}\int^\infty_{s_1}L(Q)dt\le 2{\overline a\over\underline
a}\varphi (\rho )
\end{equation}
while by (a$_2$)(ii),
\begin{equation} \label{2.21}
-\int^{s_1}_{t_1}(a(t)-a(t+\theta_2))W(Q)dt\ge \gamma\int^{s_1}_{t_1}
W(Q)\,dt\,.
\end{equation}
In the interval $[t_1,s_1]$, $Q$ goes from $\partial B_\rho (0)$ to
$\partial B_\rho (\xi )$.  In particular, since $Q$ minimizes (\ref{2.1}),
there
is a subinterval $[\sigma,
s]$ of $[t_1, s_1]$ in which $Q$ lies in ${\mathbb R}^n\backslash B_r({\mathbb Z}^n)$ and
joins $\partial B_r(0)$ to $\partial B_r(\xi )$.  Hence by the definition
of $M$,
\begin{eqnarray*}
{1\over 2}&\le& |Q(s)-Q(\sigma )|=\Big|\int^s_\sigma \dot
Q(t)dt\Big| \\
&\le& (s-\sigma )^{1/2} \Big(\int^s_\sigma |\dot Q|^2dt\Big)^{1/2} \\
&\le& (s-\sigma )^{1/2} (2I(Q))^{1/2}\\
&\le& (2M(s-\sigma ))^{1/2}
\end{eqnarray*}
so that $s-\sigma \ge 1/(8M)$, and
$$ 
-\int^{s_1}_{t_1} W(Q)dt\ge -\int^s_\sigma  W(Q)dt\ge {1\over 8M}\beta
(r)\,. 
$$
Combining (\ref{2.19})--(\ref{2.21}) and the above equation yields
$$
0\ge \gamma {1\over 8M} \beta (r)-4{\overline a\over\underline a}\varphi
(\rho )
$$
contrary to (\ref{2.15}). Hence it is not possible that $Q(t)\in\partial B_r(0)$
for $t\in (t_1, m_1]$.  Similarly using (a$_2$) (i), $Q(t)\in\partial
B_r(\xi )$ for $t\in [m_2,t_2)$ cannot occur.  Thus $Q$ is a solution of
(\ref{HS}) and Theorem \ref{t2.16} is proved.
\medskip

\begin{remark} \rm \label{r2.26} 
\begin{description}
\item{(i)} Replacing $Q_i+j$ for $j\in{\mathbb Z}^n$ gives a solution of (\ref{HS}) in
$\Gamma (m_i, m_{i+1})$ heteroclinic from $j$ to $j+\xi$.

\item{(ii)} Possibly $Q_i(\infty )\not= Q_{i-1}(\infty )$.

\item{(iii)} Although $Q_i$ need not be unique, when $a\in{\mathcal A}^*$ is
$T$-periodic, one choice for $Q_{i-1}(t)$ is $Q_i(t-T)$.
\end{description} \end{remark}

\begin{remark} \rm \label{r2.27}  Modifying slightly arguments as in 
\cite{b4, r2, b5}
gives at least $n+1$ distinct points $\xi_0\equiv\xi$, $\xi_1,\ldots
,\xi_n\in{\mathbb Z}^n\backslash\{ 0\}$ and corresponding heteroclinic solutions
$Q^0_i\equiv Q_i$, $Q^1_i,\ldots ,Q^n_i\in\Gamma (m_i, m_{i+1})$ provided
that (\ref{2.15}) is strengthened.  E.g.\ once $Q^0_i, \ldots ,Q_i^{\ell -1}$
have been found, $Q_i$ is the minimizer of the variational problem
$$
\inf_{q\in\Gamma_\ell (m_i, m_{i+1})} I(q)
$$
where
$$
\Gamma_\ell (m_i, m_{i+1})=\{ q\in\Gamma (m_i, m_{i+1})\mid q(\infty
)\not\in\hbox{span}_{\mathbb N} \{\xi_0, \ldots ,\xi_{\ell -1}\}
$$
and span$_{\mathbb N} X$ denotes the span with coefficients in ${\mathbb N}\cup \{ 0\}$ of
elements in $X$.  Moreover $Q^\ell_i$ is a solution of (\ref{HS}), $0\le i\le
n$ as in Theorem \ref{t2.16} provided that (\ref{2.15}) is replaced by
\begin{equation} \label{2.28}
\varphi (\rho )<{\gamma\over 32M^*} \big({\underline a/\overline
a}\big) \beta (r)\,,
\end{equation}
where $M^*$ is defined as follows.  Let e.g.\ $e_1,\ldots ,e_n$ be
the usual basis in ${\mathbb R}^n$, i.e.\ $e_1=(1,0,\ldots ,0)$, etc.  Set
$e_{n+1}=(-1,\ldots ,-1)$.  Replace $\eta$ in (\ref{2.5}) by $e_i$,
calling the resulting function $R_i$.  Then at least one of $R_1(1),\ldots
,R_{\ell +1}(1)\not\in \hbox{span}_{\mathbb N}\{\xi_0,\ldots ,\xi_{\ell -1}\}$.
Set
$$
M^*=\max_{1\le i\le n+1} I(R_i)\,..
$$
\end{remark}

\begin{remark} \rm \label{r2.30} As mentioned in the Introduction, the
conclusions of Theorem \ref{t2.16} hold for a more general class 
of $a$'s than ${\mathcal A}^*$. Rather than formalizing such a 
result, we just give an example of this type. Suppose 
$a=\alpha_1+\alpha_2$ where $\alpha_1\in{\mathcal A}^*$ and 
$\alpha_2\ge 0$ is continuous and periodic with period $p\le 1$ 
which for convenience will be taken to be $1$.  (Some small 
modifications in the argument that follows are needed if $p<1$.)  
It can be assumed that $\omega >>1$.  Let $\mu$ denote the 
greatest integer in $\theta_2$, $\mu =[\theta_2]$, so 
$0\le\theta_2-\mu <1\ll \omega$.  Now in the proof of Theorem 
\ref{t2.16}, choose $\tau =\mu$.  Since $\mu 
\le\theta_2<m_2-s_1-\omega +1<m_2-s_1$, $Q^*\in\Gamma$ as earlier 
so (\ref{2.18}) becomes $$ 0\ge -\int_{\mathbb R} 
(\alpha_1(t)-\alpha_1(t+\mu ))W(Q)\,dt -\int_{\mathbb R} 
(\alpha_2(t)-\alpha_2(t+\mu ))W(Q)\,dt\,. $$ By the 
$1$-periodicity of $\alpha_2$, the second integral on the right 
vanishes so the earlier argument can be used again to get 
existence here. The multiplicity results of Remark \ref{r2.27} are also 
valid for this more general class of  $a$'s. 
\end{remark}

\section{Solutions of multichain type}

Consider a heteroclinic $\ell$-chain constructed by gluing 
together $\ell$ basic heteroclinics or their translates as 
obtained in Remarks \ref{r2.27} and \ref{r2.26}(i).  Suppose the chain begins at 
$\xi_0$ and ends at $\xi_\ell$.  The goal of this section is to 
show there are infinitely many heteroclinic solutions of (\ref{HS}) that 
spend as much time as desired near $\xi_1,\ldots ,\xi_{\ell -1}$.  
To be more precise, let $r, \rho$, and ${\mathcal A}^*$ be as in 
\S2.  Let $k\in{\mathbb Z}^{2\ell}$  where $k=(k_1,\ldots ,k_{2\ell})$, 
$k_j<k_{j+1}$, and $k_j=m_{i_j}$ for some $i_j$ where $(m_i)$ is 
as in $(a_1)$. Define 
\begin{eqnarray*} 
\Gamma_k&=&\big\{ q\in 
W^{1,2}_{\rm loc}\mid q(t)\in\overline B_r(\xi_0),\; t\le k_1,\; 
q(t)\in\overline B_r(\xi_j),\; t\in [k_{2j}, k_{2j+1}],\\
&& 1\le j\le\ell -1,\hbox{ and $q(t)\in\overline B_r(\xi_\ell 
),\; t\ge k_{2\ell}$}\big\} 
\end{eqnarray*}
Set
\begin{equation} \label{3.1}
c_k=\inf_{q\in\Gamma_k} I(q)\,.
\end{equation}
Repeating arguments from \S2 gives

\begin{proposition} \label{p3.2} \begin{enumerate}

\item There exists $Q=Q_k\in\Gamma_k$ such that $I(Q_k)=c_k$.

\item There are numbers $t_1\in [k_1-\omega, k_1]$, $t_{2j}\in
[k_{2j}, k_{2j}+\omega]$, $t_{2j+1}\in [k_{2j+1}-\omega, k_{2j+1}]$, $1\le
j\le \ell -1$,
$t_{2\ell}\in [k_{2\ell}, k_{2\ell}+\omega ]$ such that $Q(t_1)\in
\overline B_\rho (\xi_0)$, $Q(t_{2j})$, $Q(t_{2j+1})\in \overline B_\rho
(\xi_j)$, $Q(t_{2\ell})\in\overline B_\rho (\xi_\ell )$.

\item There is a $\varphi (\rho )\to 0$ as $\rho\to 0$ such that
$$
\int^{t_1}_{-\infty} L(Q)dt,\;\; \int^\infty_{t_{2\ell}} L(Q)dt\le\varphi
(\rho )
$$
and similarly,
$$
\int^{t_{2j+1}}_{t_{2j}} L(Q)dt\le\varphi (\rho ),\quad 1\le j\le \ell -1
$$

\item $Q(t)\in B_{r/2}(\xi_j)$, $t\le t_1$, $Q(t)\in
B_{r/2} (\xi_j)$, $t\in [t_{2j}, t_{2j+1}]$, $Q(t)\in B_{r/2}
(\xi_\ell )$, $t\ge t_{2\ell}$.

\item There is an $s_1\in [k_1, k_1+\omega ]$, $s_{2j}\in
[k_{2j}-\omega, k]$, $s_{2j+1}\in [k_{2j+1}, k_{2j+1}+\omega ]$, $1\le
j\le
\ell -1$, $s_{2\ell}\in [k_{2\ell}-\omega, k_{2\ell}]$ such that $Q(s_1)$,
$Q(s_2)\in \overline B_\rho (\xi_0)\cup B_\rho (\xi_1), \cdots Q(s_{2\ell
-1}),
Q(s_{2\ell})\in\overline B_\rho (\xi_{\ell -1})\cup\overline B_\rho
(\xi_\ell )$.
\end{enumerate} \end{proposition}

The characterization of $Q_k$ as a minimum in (\ref{3.1}) implies there is an
$\overline M>0$ and independent of $\ell$ such that
\begin{equation} \label{3.3}
\int^{k_{2j}}_{k_{2j-1}} L(Q_k)dt\le\overline M,\quad 1\le j\le  \ell 
\end{equation}
Replacing (\ref{2.28}) by
\begin{equation} \label{3.4}
\varphi (\rho )<{\gamma\over 80\overline M} \beta (r),
\end{equation}
we have

\begin{theorem} \label{t3.5}
If (W$_1$)--(W$_2$) hold, $\rho \ll r\ll 1$, (\ref{3.4}) is 
satisfied, and $a\in{\mathcal A}^*$, then (\ref{HS}) has a solution, 
$Q_k\in\Gamma_k$ with $I(Q_k)=c_k$. 
\end{theorem}

\paragraph{Proof:}  As earlier, it suffices to show $Q(t)=Q_k(t)\not\in
\partial B_r(\xi_0)$, $t\le k_1$; $Q(t)\not\in\partial B_r(\xi_j)$, $t\in
[k_{2j}, k_{2j+1}]$, $1\le j\le \ell -1$; $Q(t)\not\in\partial
B_r(\xi_\ell)$, $t\ge k_{2\ell}$.
The idea is to show if one of these conditions is violated, it is possible
to  construct an appropriate $Q^*\in\Gamma_k$ and obtain a
contradiction as in \S2.  There are basically two cases to consider.

Suppose first that $Q(t)\in\partial B_r(\xi_0)$ for some $t\in (t_1,
k_1]$.  Then as in the proof of Theorem \ref{t2.16}, $Q(s_1)\in\overline B_\rho
(\xi_i)$, $Q(t)\in B_{r/2}(\xi_1)$ for $t\in [s_1, t_2]$, and
\begin{equation} \label{3.6}
\int^{t_2}_{s_1} L(Q)dt\le\varphi (\rho )\,.
\end{equation}
Set
\begin{equation} \label{3.7}
Q^*(t)=\left\{\begin{array}{l} 
Q(t-k_1) \qquad \mbox{if }  t\le s_1+\theta_{k_1+1}, \\[1pt] 
(s_1+\theta_{k_1+1}+1-t)Q(s_1)+(t-(s_1+\theta_{k_1+1}))\xi_1,\\
\qquad \mbox{if } s_1+\theta_{k_1+1}\le t\le s_1+\theta_{k_1+1}+1\\[2pt]
(s_1+\theta_{k_1+1}+2-t)\xi_1+(t-(s_1+\theta_{k_1+1}+1))
Q(s_1+\theta_{k_1+1}+2),\\
\qquad \mbox{if }  s_1+\theta_{k_1+1}+1\le t\le
s_1+\theta_{k_1+1}+2\\[2pt]
Q(t),\qquad\mbox{if }  t\ge s_1+\theta_{k_1+1}+2.
\end{array}\right.
\end{equation}
Then $Q^*\in\Gamma_k$ and
\begin{eqnarray*}
0\ge I(Q)-I(Q^*)&=&\int^{s_1}_{-\infty}
L(Q)dt-\int^{s_1+\theta_{k_1+1}+1}_{-\infty} L(Q^*)\,dt \\
&&+\int^{s_1+\theta_{k_1+1}+2}_{s_1}
L(Q)dt-\int^{s_1+\theta_{k_1+1}+2}_{s_1+\theta_{k_1+1}} L(Q^*)\,dt\,.
\end{eqnarray*}
By (\ref{3.6}), each of the last two terms in this inequality is  less than or
equal to $\varphi (\rho )$. Therefore,
$$
0\ge -\int^{s_1}_{-\infty} (a(t)-a(t+\theta_{k_1+1}))W(Q)dt-2\varphi (\rho).
$$
As in \S2, this leads to
$$
0\ge \gamma {1\over 8\overline M} \beta (r)-3\varphi (\rho )
$$
contrary to (\ref{3.4}).

Using (a$_2$)(i), a similar argument holds if $Q(t)\in\partial
B_r(\xi_{2\ell})$ for some $t\in [k_{2\ell}, t_{2\ell})$.
If $Q(t)\in\partial B_r(\xi_j)$ for some $t\in [k_{2j}, k_{2j+1}]$, then
by Proposition \ref{p3.2}, either $t\in [k_{2j}, t_{2j})$ or $t\in (t_{2j+1},
k_{2j+1}]$.  The argument is similar in either event, so suppose $t\in
[k_{2j}, t_{2j})$.  Then $Q(s_{2j})\in\overline B_\rho (\xi_{j-1})$ and
$Q(t)\in B_{r/2}(\xi_{j-1})$ for $t\in [t_{2j-1}, s_{2j}]$.  It is
now convenient to use two comparison functions.  Define
$$
 \widetilde Q(t)=\cases{
Q(t)      & if $t\le t_{2j-1}$ \cr
\xi_{j-1} & if $t_{2j-1}+1\le t\le s_{2j}-1$ \cr
Q(t)      & if $s_{2j}\le t\le t_{2j}$ \cr
\xi_j     & if $t_{2j+1}\le t\le t_{2j+1}-1$ \cr
Q(t)      & if $t\ge t_{2j}$ 
} $$ 
with a linear interpolant, as in (\ref{3.7}) for the four intermediate
intervals.  Then $\widetilde Q\in\Gamma_k$ and
\begin{eqnarray*}
0&\le& I(\widetilde Q)-I(Q)\\
&=&\int^{t_{2j-1}+1}_{t_{2j-1}}L(\widetilde
Q)dt+\int^{s_{2j}}_{s_{2j-1}}L(\widetilde Q)\,dt 
+\int^{t_{2j+1}}_{t_{2j}} L(\widetilde Q)\,dt \\
&&+\int^{t_{2j+1}}_{t_{2j+1}-1} L(\widetilde Q)\,dt  
-\int^{s_{2j}}_{t_{2j-1}} L(Q)\,dt-\int^{t_{2j+1}}_{t_{2j}}L(Q)\,dt\,.
\end{eqnarray*}
Each of the terms on the right-hand side of this inequality is less than
or equal to $\varphi (\rho)$ so
\begin{equation} \label{3.13}
0\le I(\widetilde Q)-I(Q)\le 6\varphi (\rho )\,.
\end{equation}
Now define
$$
Q^*(t)=\cases{ \widetilde Q(t) & if $t\le t_{2j}+1-\theta_j$\cr
 \widetilde Q(t+\theta_j)  & if $t_{2j+1}+1-\theta_j\le t\le t_{2j}+1$\cr
\widetilde Q(t)            & if $t\ge t_{2j}+1$\,. }
$$
Again $Q^*\in\Gamma_k$ and
\begin{equation} \label{3.15}
0\le I(Q^*)-I(Q)=I(Q^*)-I(\widetilde Q)+I(\widetilde Q)-I(Q)\,.
\end{equation}
Hence by (3.13),
\begin{equation} \label{3.16}
I(\widetilde Q)-I(Q^*)\le I(\widetilde Q)-I(Q)\le 6\varphi (\rho)\,.
\end{equation}
But by the definition of $Q^*$ and $\widetilde Q$,
\begin{eqnarray}
I(\widetilde Q)-I(Q^*)&=&\int^{t_{2j}+1}_{t_{2j-1}+1} (L(\widetilde
Q)-L(Q^*))dt\nonumber \\
 &=&-\int^{t_{2j+1}}_{s_{2j-1}}(a(t)-a(t-\theta_j))W(Q)dt \label{3.17}\\
&\ge& -\int^{t_{2j}}_{s_{2j}}(a(t)-a(t-\theta_j))W(Q)dt-4\varphi (\rho )
\nonumber\\
&\ge& {\gamma \over 8\overline M}\beta (r)-4\varphi (\rho ). \nonumber
\end{eqnarray}
Combining (\ref{3.15})--(\ref{3.17}) shows
$$
{\gamma \beta (r)\over 8\overline M}\le 10\varphi (\rho )
$$
contrary to (\ref{3.4}).  The proof is complete.


\begin{remark} \rm \label{r3.19}  By choosing $k$ appropriately, the solution, $Q_k$,
of (\ref{HS})  is near each of the equilibrium points $\xi_1,\ldots ,\xi_{\ell
-1}$ for as long a time interval as desired.  However $Q_k$ need not be
near the original heteroclinic chain joining $0$ and $\xi_\ell$, i.e.
$Q_k\big|^{k_{2j}}_{k_{2j-1}}$ is not necessarily near any basic
heteroclinic joining $\xi_{j-1}$ and $\xi_j$.  Nevertheless, a
$Q_k\big|^{k_{2j}}_{k_{2j-1}}$ near such $P_j$ can be constructed by
taking $k_{2j}-k_{2j-1}$ sufficiently large as in \cite{r4}.  Indeed (\ref{3.3})
implies an $L^\infty$ upper bound for $Q_k\big|^{k_{2j}}_{k_{2j-1}}$
independent of $k_{2j}-k_{2j-1}$ and (\ref{HS}) then yields  such a bound in
$C^2$.
As $k_{2j}-k_{2j-1}\to\infty$, by standard arguments as in \cite{r4},
$Q_k\big|^{k_{2j}}_{k_{2j-1}}$ approaches a chain of heteroclinic
$H,\ldots ,H_s$ joining $\xi_{j-1}$ and $\xi_j$ with
$$
\sum^s_1 I(H_i)=I(P_j).
$$
The construction of $P_j$ as indicated in Remark \ref{r2.27} implies $s=1$.
Hence for $k_{2j}-k_{2j-1}$ large, $Q_k\big|^{k_{2j}}_{k_{2j-1}}$ will be
near a basic heteroclinic $P_j$ joining $\xi_{j-1}$ and $\xi_j$.
\end{remark}

A standard consequence of Theorem \ref{t3.5} is the existence of solutions of
infinite chain type of (\ref{HS}).  Consider any formal doubly infinite
heteroclinic chain made up of the basic heteroclinics of Remark
\ref{r2.27}. 
 The
endpoints of the chain form a sequence $\Xi =(\xi_i)_{i\in{\mathbb Z}}$,
$\xi_i\in{\mathbb Z}^n$.  Let $k=(k_i)_{i\in{\mathbb Z}}$ with $k_i<k_{i+1}$ and each
$k_i=m_{i_j}$ for some $j$.  Now set
$$
\Gamma_k=\{ q\in W^{1,2}_{\rm loc}\mid q(t)\in\overline B_r(\xi_j),\; t\in
[k_{2j}, k_{2j+1}],\; j\in{\mathbb Z}\}.
$$
Then we have

\begin{theorem} \label{t3.20}
Under the hypothesis of Theorem \ref{t3.5}, for each
$\Xi$, $k$ as above, there is a solution, $Q_k\in\Gamma_k$, of (\ref{HS}).
\end{theorem}

\paragraph{Proof:}  Note that the construction of Theorem \ref{t3.5} is 
independent of $\ell$, the number of basic homoclinics.  For $\Xi$ and $k$ as
above, let $\Xi_\ell =(\xi_{-\ell},\ldots ,\xi_\ell )$ and $K_\ell
=(k_{-2\ell},\ldots ,k_{2\ell})\in {\mathbb Z}^{4\ell}$.  Then by Theorem \ref{t3.5},
there is a solution $Q_\ell$ of (\ref{HS}) in $\Gamma_{K_\ell}$, heteroclinic
from $\xi_{-\ell}$ to $\xi_\ell$.  Since $Q_\ell$ is a solution of (\ref{HS}),
for each $j\in{\mathbb Z}$, the form of $\Gamma_{K_\ell}$ yields $C^2([k_{2j},
k_{2j+1}], {\mathbb R}^n)$ bounds for $Q_\ell$ (independent of $\ell$).  Moreover
in the intervals $[k_{2j-1}, k_{2j}]$, the bound (\ref{3.3}) holds with
$Q=Q_\ell$.  As in Remark \ref{r3.19}, this gives bounds in
$C^2([k_{2j-1},
k_{2j}], {\mathbb R}^n)$ for $Q_\ell$ independent of $\ell$.  The Arzela-Ascoli
Theorem then yields the desired solution $Q_k$ of (\ref{HS}).

\paragraph{Acknowledgement}
Coti Zelati's research was done while he was visiting the
University of Wisconsin--Madison.  He thanks the faculty and staff of the
Mathematics Department for their kind hospitality. 

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\noindent{\sc Paul H. Rabinowitz} \\ 
Department of Mathematics \\
University of Wisconsin-Madison \\
Madison, WI 53706, USA \\
e-mail: rabinowi@math.wisc.edu\medskip


\noindent{\sc Vittorio Coti Zelati} \\
Departimento di Matematica \\
Universita di Napoli \\
Napoli, Italy \\ 
e-mail: zelati@unina.it
 \end{document}