The critical dimension for a fourth order elliptic problem with singular nonlinearity

Abstract

We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\fλ{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary conditions $u={\partial }_{\nu }u=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" $\la^*>0$ such that a stable classical solution $u_\la$ with $0<u_\la<1$ exists for $\la\in (0,\la^*)$, while there is none of any kind when $\la>\la^*$. Our main result asserts that the extremal solution $u_{{\lambda }^{\ast }}$ is regular $({\sup }_B{u}_{{\lambda }^{\ast }}<1)$ provided $N\le 8$ while $u_{{\lambda }^{\ast }}$ is singular (${\sup }_B{u}_{{\lambda }^{\ast }}=1$) for $N\ge 17$, in which case $1-C_0|x{|}^{4/3}\le u_{{\lambda }^{\ast }}(x)\le 1-|x{|}^{4/3}$ on the unit ball, where $C_0:=<(\frac{{\lambda }^{\ast }}{\bar{\lambda }}>{)}^{1/3}$ and $\bar{\lambda }:=\frac{8(N-2/3)(N-8/3)}{9}$. The singular character of the extremal solution for the remaining cases (i.e., when $9\le N\le 16$) requires a computer assisted proof and will not be addressed in this paper.

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