On semi-linear elliptic equation arising from Micro-Electromechanical Systems with contacting elastic membrane

Abstract

This paper is concerned with the nonlinear elliptic problem $-\Delta u=\frac{\lambda }{(a-u{)}^2}$ on a bounded domain $\Omega$ of ${\mathbb{R}}^N$ with Dirichlet boundary conditions. This problem arises from Micro-Electromechanical Systems devices in the case that the elastic membrane contacts the ground plate on the boundary. We analyze the properties of minimal solutions to this equation when $\lambda >0$ and the function $a:\bar{\Omega }\to [0,1]$ satisfying $a(x)\ge \kappa \mathrm{dist}(x,\partial \Omega {)}^{\gamma }$ for some $\kappa >0$ and $\gamma \in (0,1)$. Our results show how the boundary decay of the membrane works on the solutions and pull-in voltage $\lambda$.

18 pages. arXiv admin note: text overlap with arXiv:math/0509534, arXiv:0712.3071 by other authors