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

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 $\lambda>0$, $N\geq 1$, $p>0$ and $a:\bar\Omega\to[0,1]$ vanishes at the boundary with the rate ${\rm dist}(x,\partial\Omega)^\gamma$ 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.