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

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 $a(x)\ge \kappa{\rm dist}(x,\partial\Omega)^\gamma$ 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 $\lambda$.