On semilinear elliptic equation with negative exponent arising from a closed MEMS model

Abstract

This paper is concerned with the elliptic equation $-\Delta u=\frac{\lambda }{(a-u{)}^p}$ in a connected, bounded $C^2$ domain $\Omega$ of ${\mathbb{R}}^N$ subject to zero Dirichlet boundary conditions, where $\lambda >0$, $N\ge 1$, $p>0$ and $a:\bar{\Omega }\to [0,1]$ vanishes at the boundary with the rate $\mathrm{dist}(x,\partial \Omega {)}^{\gamma }$ for $\gamma >0$. When $p=2$ and $N=2$, this equation models the closed Micro-Electromechanical Systems devices, where the elastic membrane sticks the curved ground plate on the boundary, but insulating on the boundary. The function $a$ shapes the curved ground plate. Our aim in this paper is to study qualitative properties of minimal solutions of this equation when $\lambda >0$, $p>0$ and to show how the boundary decaying of $a$ works on the minimal solutions and the pull-in voltage. Particularly, we give a complete analysis for the stability of the minimal solutions.

24 pages, 3 figures