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

This paper is concerned with the nonlinear elliptic problem $-Δu=\frac{λ}{(a-u)^2}$ on a bounded domain $Ω$ 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 $λ>0$ and the function $a:\barΩ\to[0,1]$ satisfying $a(x)\ge κ{\rm dist}(x,\partialΩ)^γ$ for some $κ>0$ and $γ\in(0,1)$. Our results show how the boundary decay of the membrane works on the solutions and pull-in voltage $λ$.

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