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

This paper is concerned with the nonlinear elliptic problem $-\Delta u=\frac{\lambda }{(a-u)^2}$onaboundeddomain$\Omega$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 $a:\bar{\mathrm{\Omega }}\to [0,1]$ satisfying $a(x)\ge \kappa \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathrm{\partial }\mathrm{\Omega }{)}^{\gamma }$ 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 $\lambda$.