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\markboth{\hfil A Dirichlet problem in the strip \hfil EJDE--1996/10}%
{EJDE--1996/10\hfil Eugenio Montefusco \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1996}(1996), No.\ 10, pp. 1--9. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\vspace{\bigskipamount} \\
A Dirichlet problem in the strip
\thanks{ {\em 1991 Mathematics Subject Classifications:}
35J65, 35B50.\newline\indent
{\em Key words and phrases:} Maximum Principle, Sliding Method,
\newline\indent Subsolution and Supersolution.
\newline\indent
\copyright 1996 Southwest Texas State University and University of
North Texas.\newline\indent
Submitted June 8, 1996. Published October 26, 1996.} }
\date{}
\author{Eugenio Montefusco}
\maketitle
\begin{abstract}
In this paper we investigate a Dirichlet problem in a strip and, using
the sliding method, we prove monotonicity for positive and bounded
solutions. We obtain uniqueness of the solution and show that this
solution is a function of only one variable.
From these qualitative properties we deduce existence of a classical
solution for this problem.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\section{Introduction}
In 1979 B. Gidas, W. M. Ni and L. Nirenberg studied the problem:
\begin{equation}
\label{Ginini}\left\{
\begin{array}{cl}
-\Delta u=f(u) & \mbox{in }B(0,r) \\ u\equiv 0 & \mbox{on }\partial B(0,r)
\end{array}
\right.
\end{equation}
where $f$ is a locally Lipschitz function. In \cite{GNN} they showed that
the solution of (\ref{Ginini}) is a radial function, therefore this solution
reflects the symmetry of the domain. The proof of this result is based on
the moving plane method and the maximum Principle.
In the last years the interest in qualitative properties of
solutions of nonlinear elliptic equations has increased.
H. Berestycki and L. Nirenberg \cite{BN} have simplified the moving plane
method and proved the symmetry of solutions of elliptic equations in
nonsmooth domains. In the
same paper H. Berestycki and L. Nirenberg have also simplified the sliding
method, which is a technique for proving monotonicity of solutions of
nonlinear elliptic equations.
At the same time some mathematicians are interested in qualitative
properties of solutions of elliptic equations in unbounded domains. H.
Berestycki and L. Nirenberg have studied the flame propagation in
cylindrical domains \cite{BN2}. C. Li investigated elliptic equations
in various unbounded domains \cite{L}. H. Berestycki, M. Grossi and F.
Pacella showed (using the moving plane method) that an equation with
critical growth does not admit a solution in the half space \cite{BGP}.
In 1993, H. Berestycki, L. A. Caffarelli and L. Nirenberg considered a
Dirichlet problem in the half space, they showed that the solution is a
function of only one variable (under suitable hypotheses) using the sliding
method \cite{BCN}.
With the same technique we want to prove a similar result in the strip. In
fact, using the sliding method, we show that problem (\ref{striscia2}) has a
unique classical solution depending on one variable only. As a matter
of fact, the problem is reduced to an ODE.
This paper is organized as follows. In section 2, we study the
qualitative properties of the solution to (\ref{striscia2}).
In section 3 we show some simple corollaries to the qualitative study.
In this paper we use frequently the following two theorems.
\begin{theorem}
Let $\Omega $ be an arbitrary bounded domain of $\Bbb R^N$ which is convex
in the $x_1$-direction. Let $u\in W_{loc}^{2,N}\left( \Omega \right) \cap
C\left( \overline{\Omega }\right) $ be a solution of:
\begin{equation}
\label{sliding}\left\{
\begin{array}{cl}
\Delta u+f(u)=0 & \mbox{in }\Omega \,, \\ u\equiv \varphi & \mbox{on
}\partial \Omega \,.
\end{array}
\right.
\end{equation}
The function $f$ is assumed to be Lipschitz continuous. Here we assume that
for any three points $x'=(x'_1,y)\,$, $x=(x,y)\,$, $%
x'' =(x''_1,y)$ lying on a segment parallel to
the $x_1$-axis, $x'_1 u(x_1,y)\quad \mbox{for }(x_1,y)\,,\,(x_1+\tau ,y)\in \Omega
\;\mbox{and }\tau >0 .
$$
Furthermore, if $f$ is differentiable, then $u_{x_1}>0$ in $\Omega $.
Finally, $u$ is the unique solution of (\ref{sliding}) in $%
W_{\small\mbox{loc}}^{2,N}(\Omega )\cap C\left( \overline{\Omega }
\right) $ satisfying (\ref{monotonia}).
\end{theorem}
\paragraph{Proof.} See Theorem 1.4 of \cite{BN}.
\begin{theorem}
Let $\Omega$ be a bounded domain and
suppose $u_1\in H^1\left( \Omega \right) $ is a subsolution while $u_2\in
H^1\left( \Omega \right) $ is a supersolution to problem (\ref{sliding}),
let be $f\in C( \Bbb R) $ and assume that with constants $%
c_1\,,\,c_2\in \Bbb R$ there holds $-\infty 0$ in $S$. Moreover, $u$ is the unique
solution of (\ref{striscia2}).
\end{theorem}
The proof of Theorem 2.1 relies on the following propositions:
\begin{proposition}
There exists $w(t)$, a solution of
\begin{equation}
\label{sopra}\left\{
\begin{array}{cc}
w''(t)+f(w(t))=0 & \mbox{in }(0,h) \\ 00$ in $S$ and $u$ is the unique solution of (%
\ref{striscia2}).
Now we must prove Propositions 2.2 and 2.3, which are more difficult than
Theorem 2.1.
\subsection*{Proof of Proposition 2.2}
Since the function $u$ is a classical solution of (\ref{striscia2}),
by the Schauder estimates we can say that $|\nabla u|\leq K$ in the
strip, with $K$
depending only on $\max _{[0,h]}f(s)$, (see Theorem 8.33 in \cite{GT}).
For $\varepsilon \in \left( 0,\min \left[1/K,h/M\right]\right)$,
consider the function on $\Bbb R^{+}$:
$$
\sigma _\varepsilon (t)=\left\{
\begin{array}{cc}
\frac t\varepsilon & \mbox{in }[0,\varepsilon M] \\ M & \mbox{in }[%
\varepsilon M,h].
\end{array}
\right.
$$
Set $\Omega _R:=\left\{ x\in S:\left( x_2^2+\dots +x_N^2\right) ^{1/2}
R$. For $a\in \Bbb R^{N-1}$, such that $%
|a|w_{R^{\prime }}(x_1,x^{\prime }+a)\quad \forall a\in
\Bbb R^{N-1}\mbox{, with }|a|0$ (using also the strong
maximum Principle).
We want to slide the translated $\Omega _R$ by increasing $\delta $, and use
the maximum Principle to show that $\rho _\delta >0$ in $\Sigma _\delta $
for every positive $\delta 0$ in $\Sigma
_\delta $ for a maximal open interval $(0,\mu )$, with $\mu \le \delta $.
We want to show that $\mu =h$ by contradiction. Assume that $\mu 0$ on $\{x_1=0\}$ we have that $\rho _\mu \not
\equiv 0$; therefore, by the maximum Principle, we can say that $\rho _\mu
>0 $ in $\Sigma _\mu $.
We choose a small positive real number $\alpha $ such that $\alpha <\min
[(h-\mu ),R]$, and consider the subset $A:=\{(x,x^{\prime })\in \Omega
_R:x_1<(\mu -\alpha )\,,\,|x^{\prime }|<(R-\alpha )\}$.
As $\rho _\mu >0$ in $\Sigma _\mu $, there exists some constant $\varepsilon
>0$ such that $\rho _\mu \ge \varepsilon $ in $\overline{A}$. Thus for $\mu
^{\prime }>\mu $, with $(\mu ^{\prime }-\mu )$ sufficiently small, we obtain
that $\rho _{\mu ^{\prime }}>0$ in $\overline{A}$.
To conclude that $\rho _{\mu ^{\prime }}>0$ in $\Sigma _{\mu ^{\prime }}$ we
use the maximum Principle again. In $\Sigma _{\mu ^{\prime }}\setminus
\overline{A}$ the function $\rho _{\mu ^{\prime }}$ verifies $-\Delta \rho
_{\mu ^{\prime }}=c(x)\rho _{\mu ^{\prime }}$, and since $\rho _{\mu
^{\prime }}>0$ in $\overline{A}$, we also have $\lim \inf _{x\rightarrow
\partial (\Sigma _{\mu ^{\prime }}\setminus \overline{A})}\rho _{\mu
^{\prime }}(x)\ge 0$. Since $\left( \Sigma _{\mu ^{\prime }}\setminus
\overline{A}\right) $ has small measure for $(\mu ^{\prime }-\mu )$ small,
the maximum principle holds in $\left( \Sigma _{\mu ^{\prime }}\setminus
\overline{A}\right) $ (see Proposition 1.1 in \cite{BN}), and we conclude
that $\rho _{\mu ^{\prime }}>0$ in $\left( \Sigma _{\mu ^{\prime }}\setminus
\overline{A}\right) $, and hence in all of $\Sigma _{\mu ^{\prime }}$. This
is impossible for the maximality of $(0,\mu )$, therefore we have proved
that $\rho _\delta >0$ in $\Sigma _\delta ,\;\forall \delta R$. For $a\in \Bbb R^{N-1}$ such that $%
|a|<\left( R^{^{\prime }}-R\right) $, and for $\delta \in (0,h)$, we slide $%
\Omega _R$ so that its center is at $\left( a,\delta -\frac h2\right) $. For
$\delta $ small, by continuity, the translated $\rho _R$ is less than $\rho
_{R^{\prime }}$ in the overlapped region with $\Omega _{R^{\prime }}$.
Moving the displaced $\Omega _R$, and using the sliding method, we conclude
that
\begin{equation}
\label{disug2}\rho _{R^{\prime }}\left( x\right) >\rho _R(x_1,x^{\prime
}+a)\quad \forall a\in \Bbb R^{N-1}\mbox{, with }|a|