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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 101, pp. 1--4.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/101\hfil Elliptic equations in large dimensions]
{Solvability of quasilinear elliptic equations in large dimensions}
\author[O. Zubelevich\hfil EJDE-2005/101\hfilneg]
{Oleg Zubelevich}

\address{Oleg Zubelevich \hfill\break
Department of Differential Equations and Mathematical Physics\\
Peoples Friendship University of  Russia\\
Ordzhonikidze st., 3,  117198,  Moscow, Russia}
\email{ozubel@yandex.ru}
\curraddr{2-nd  Krestovskii Pereulok 12-179, 129110, Moscow, Russia}

\date{}
\thanks{Submitted September 8, 2005. Published September 21, 2005.}
\thanks{Partially supported by grant RFBR 05-01-01119.}
\subjclass[2000]{35J60}
\keywords{Boundary-value problems; nonlinear elliptic equations}

\begin{abstract}
 We study the solvability of quasilinear elliptic Dirchlet
 boundary-value  problems.  In particular, we show that if the
 dimension of the domain is large enough  then the solution
 exists independent of the growth rate on right-hand side.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

It is well known the boundary-value problem
$$
-\Delta u=|u|^{p-2}u,\quad u\big|_{\partial M}=0,
$$
where $M$ is an $m$-dimensional star-shaped bounded domain,
has nontrivial solutions in $H^1_0(M)$ provided that $p<2m/(m-2)$;
see for example \cite{Rabinowicz}.
It is also known that by the Pohozaev's identity, if $p>2m/(m-2)$
there is no non-trivial solution; see for example \cite{Pohozaev}.
This indicates the importance of the growth rate of the right-hand
side.

On the other hand, the dimension of the domain $M$ also plays a role on
the existence of solutions. In this note, we show that if the dimension
of the domain is large enough then the solution exists independent
of the growth rate on the right-hand side.

\section{Main theorem}

Let $M$ be a bounded domain in $\mathbb{R}^m$ with smooth boundary
$\partial M$. For $x=(x_1,\ldots,x_m)$, we use
the standard Euclidian norm $|x|^2=\sum_{i=1}^{m}x_i^2$.
We assume that the domain $M$ is contained in a ball of radius $R$:
$$
M\subseteq B_R(x_0,\mathbb{R}^m)=\{x\in\mathbb{R}^m: |x-x_0|<R\}.
$$
We use the Banach space
$$
C_0^1(\overline M)=\{v\in  C^1(\overline M): v\big|_{\partial M}=0\}.
$$
For the right-hand side, we use function
$f:C_0^1(\overline M)\to L^\infty(M)$ which is continuous.
The our main objective is to show the existence of solutions to
elliptic problem
\begin{equation}
\label{main_eq}
-\Delta u=f(u),\quad u\big|_{\partial M}=0.
\end{equation}

\begin{theorem} \label{main_t}
Suppose there exists a constant $\lambda$ such that for any
$v\in C_0^1(\overline M)$ with $|v(x)|\le \lambda$,
the inequality
\begin{equation} \label{th_f}
|f(v)|\le \frac{2m\lambda}{R^2}\end{equation}
 holds almost everywhere (a.e.) in $M$.
Then problem (\ref{main_eq}) has a solution
$$
u\in \widetilde{H}^{2,r}(M):=H^{1,r}_0(M)\bigcap H^{2,r}(M),\quad r>m.
$$
\end{theorem}

As an example of a right-hand side that satisfies the conditions
above, we have $f(u)=(2+\cos(|\nabla u|^2))e^u$.

Let $M_m\subset \mathbb{R}^m,$ be a sequence of bounded domains with
smooth boundaries and inscribed in Euclidian balls with a given radius $R$.
Let $g:\mathbb{R}\to \mathbb{R}$ be a continuous function and $f$ be
mapping $f(v)=g(v(x))$.

Consider problem (\ref{main_eq}) with  $f$ defined on the domains $M_m$.
We claim that in such case  problem (\ref{main_eq}) has a solution
provided $m$ is sufficiently large. Indeed, take a positive
constant $\lambda$ and observe that the function $g$ is bounded in the
closed interval $[-\lambda,\lambda]$, thus inequality (\ref{th_f}) will
certainly be fulfilled, when the number $m$ is sufficiently large.

To illustrate this effect consider the example:
\begin{equation}
\label{prim1}
-\Delta u=ce^u,\quad u\big|_{\partial B_1(0,\mathbb{R}^m)}=0,
\end{equation}
where $c$ is a positive constant.

In the one-dimensional case, equation (\ref{prim1}) can be integrated
explicitly. However the corresponding integrals can not be expressed  by
elementary functions. Numerical simulation of these integrals shows that
the problem (\ref{prim1}) has a solution if and only if
$$
c\le 0,87845\dots
$$
On the other hand, applying Theorem \ref{main_t} with
 $|v(x)|\le \lambda$ one has:
\begin{equation}
\label{eq_es}
ce^v\le ce^\lambda \le 2m\lambda.\end{equation}
If
$$
c\le 2e^{-1}m= m\cdot 0.73575\dots
$$
then the second inequality of (\ref{eq_es}) has a solution $\lambda$.
So letting $c=1$, we see that problem (\ref{prim1}) has no
solutions in the one-dimensional case, and by Theorem \ref{main_t}
it has a solution for $m\ge 2$.

To conclude, we note that by Proposition \ref{max_p} (see below),
the solution to (\ref{prim1}) is nonnegative.

\section{Proof of Main Theorem}

The arguments presented here are quite standard: We use a version of the
comparison principle.
Denote by $\Delta^{-1}h$ the solution of the problem
$$
\Delta w=h\in H^{s,p}(M),\quad w\big|_{\partial M}=0,\quad s\ge 0, \; p>1.
$$
It is well known that the linear mapping $\Delta^{-1}:H^{s,p}(M)\to
\widetilde{H}^{s+2,p}(M)$ is bounded.
Now we construct a mapping
$$
G(v)=-\Delta^{-1}f(v)
$$
and look for a fixed point of this mapping.

By the assumptions above, $G:C_0^1(\overline M)\to \widetilde{H}^{2,r}(M)$ is
continuous and by virtue of the embeddings:
\begin{equation}
\label{incl}
\widetilde{H}^{2,r}(M)\sqsubset \widetilde{H}^{2-\delta,r}(M)\subset
C_0^1(\overline M),\quad 0<\delta<1,\; (1-\delta)r>m,
\end{equation}
(here $\sqsubset$ is a completely continuous embedding)
the mapping  $G:C_0^1(\overline M)\to C_0^1(\overline M)$ is completely
continuous.

Consider a function
$$
U(x)=\frac{\lambda}{R^2}(R^2-|x-x_0|^2).
$$
This function takes positive values for $x\in B_R(x_0,\mathbb{R}^m)$, attains its
maximum at $x_0$:
$$
\max_{B_R(x_0,\mathbb{R}^m)}U=U(x_0)=\lambda,$$
and satisfy the  Poisson equation
\begin{equation}
\label{U_P}
-\Delta U=\frac{2m\lambda}{R^2}.
\end{equation}
Let us recall a version of the maximum principle.

\begin{proposition}[\cite{Taylor1}] \label{max_p}
IF $v\in H^1(M)$ and
$\Delta v\ge 0$ then inequality $v(x)\le 0$ a.e. in $\partial M$
implies that
$v(x)\le 0$ a.e. in $M$.
\end{proposition}

\begin{lemma} \label{main_lem}
The function $G$ maps the set
$$
W=\{w\in C_0^1(\overline M)\mid |w(x)|\le \lambda,\quad x\in M\}
$$
to itself. Furthermore, the set $G(W)$ is bounded in $\widetilde{H}^{2,r}(M)$.
\end{lemma}

\begin{proof}
Since $-\Delta G(w)=f(w)$, by formula (\ref{U_P}) one has
$$
\Delta(G(w)-U)=-f(w)+\frac{2m\lambda}{R^2}\ge 0
$$
a.e. in $M$. Observing that
$(G(w)-U)\big|_{\partial M}=-U\big|_{\partial M}\le 0$,
 by Proposition \ref{max_p} we see that $G(w)\le U$ a.e. in $M$.
The same arguments give $-U\le G(w)$ a.e. in $M$.
Note that, a.e. in $M$, we have
$$
|G(w)|\le U\le \max_{B_R(x_0,\mathbb{R}^m)}U=\lambda.
$$
By assumption of this Theorem, the set $f(W)$ is bounded in $L^\infty(M)$;
i.e., $|f(W)|\le2m\lambda/R^2$.  Consequently the set $\Delta^{-1} f(W)$
is bounded in $\widetilde{H}^{2,r}(M)$.
\end{proof}

Note that Lemma \ref{main_lem} and formula (\ref{incl}) imply the
 set $G(W)$ being precompact in $C_0^1(\overline M)$. Observing that $W$
is a convex set, we apply Schauder's fixed point theorem to the mapping
$G:W\to W$ and obtain desired fixed point $u=G(u)\in \widetilde{H}^{2,r}(M)$.
This completes the proof of the main  Theorem.

\begin{thebibliography}{0}
\bibitem{Pohozaev} 
S. I. Pohozaev; \emph{On the eigenfunctions of the
equation $\Delta u+\lambda f(u)=0,$} Soviet Math. Dokl. 6 (1965), 1408-1411.
\bibitem{Rabinowicz} 
P. H. Rabinowicz;
\emph{Minimax Methods in Critical Point Theory with
Applications to Differential Equations}, Regional Conference
Series in Mathematics, American Mathematical Society, 1986.
\bibitem{Taylor1}
 M. E. Taylor;
\emph{Partial Differential Equations}, Vol. 1, Springer, New York, 1996.


\end{thebibliography}
\end{document}