\documentclass[reqno]{amsart} \usepackage{hyperref} \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 $R$: $$M\subseteq B_R(x_0,\mathbb{R}^m)=\{x\in\mathbb{R}^m: |x-x_0|m.$$ \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: $$\label{prim1} -\Delta u=ce^u,\quad u\big|_{\partial B_1(0,\mathbb{R}^m)}=0,$$ 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: $$\label{eq_es} ce^v\le ce^\lambda \le 2m\lambda.$$ 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: $$\label{incl} \widetilde{H}^{2,r}(M)\sqsubset \widetilde{H}^{2-\delta,r}(M)\subset C_0^1(\overline M),\quad 0<\delta<1,\; (1-\delta)r>m,$$ (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 $$\label{U_P} -\Delta U=\frac{2m\lambda}{R^2}.$$ 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}