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 $u=\partial_νu=0$ 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 $0 \la^*$. Our main result asserts that the extremal solution $u_{λ^*}$ is regular $(\sup_B u_{λ^*} <1)$ provided $ N \le 8$ while $u_{λ^*} $ is singular ($\sup_B u_{λ^*} =1$) for $N \ge 17$, in which case $1-C_0|x|^{4/3}\leq u_{λ^*} (x) \leq 1-|x|^{4/3}$ on the unit ball, where $ C_0:= <(\frac{λ^*}{\barλ}>)^{1/3}$ and $ \barλ:= \frac{8 (N-{2/3}) (N- {8/3})}{9}$. The singular character of the extremal solution for the remaining cases (i.e., when $9\leq N\leq 16$) requires a computer assisted proof and will not be addressed in this paper.