The critical dimension for a 4th order problem with singular nonlinearity

TL;DR

This paper investigates the regularity of extremal solutions to a singular nonlinear biharmonic equation modeling MEMS devices, identifying a critical dimension at N=8 where solutions transition from regular to singular.

Contribution

It completes previous results by precisely characterizing the extremal solution's regularity based on the spatial dimension, establishing the critical dimension at N=8.

Findings

Extremal solution is regular for N ≤ 8.

Extremal solution is singular for N ≥ 9.

Explicit bounds for the extremal solution near the singularity.

Abstract

We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\f{\lambda}{(1-u)^2}$, 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 $u_{\lambda^*}$ is regular $(\sup_B u_{\lambda^*} <1)$ provided $ N \le 8$ while $u_{\lambda^*} $ is singular ($\sup_B u_{\lambda^*} =1$) for $N \ge 9$, 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^*}{\overline{\lambda}})^{1/3}$ and $ \bar{\lambda}:= {8/9} (N-{2/3}) (N- {8/3})$.