# The critical dimension for a fourth order elliptic problem with singular nonlinearity - Mathematics > Analysis of PDEs

Abstract: We study the regularity of the extremal solution of the semilinear biharmonicequation $\bi u=\f{\lambda}{1-u^2}$, which models a simpleMicro-Electromechanical System MEMS device on a ball $B\subset\IR^N$, underDirichlet boundary conditions $u=\partial u u=0$ on $\partial B$. We completehere the results of F.H. Lin and Y.S. Yang \cite{LY} regarding theidentification of a -pull-in voltage- $\la^*>0$ such that a stable classicalsolution $u \la$ with $0\la^*$. Our main result asserts that the extremalsolution $u {\lambda^*}$ is regular $\sup B u {\lambda^*} <1$ provided $N\le 8$ while $u {\lambda^*}$ is singular $\sup B u {\lambda^*} =1$ for $N\ge 17$, in which case $1-C 0|x|^{4-3}\leq u {\lambda^*} x \leq 1-|x|^{4-3}$on the unit ball, where $C 0:= <\frac{\lambda^*}{\bar{\lambda}}>^{1-3}$ and$\bar{\lambda}:= \frac{8 N-{2-3} N- {8-3}}{9}$. The singular character ofthe 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.

Author: Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub

Source: https://arxiv.org/