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

This paper is concerned with the elliptic equation -Δu=λ(a-u)p in a connected, bounded C² domain Ω of R^N subject to zero Dirichlet boundary conditions, where λ> 0 , p> 0 and a: Ω ¯ → [0, 1] vanishes at the boundary with the rate dist (x, ∂Ω ) ^γ for γ> 0 . When N= 2 and p= 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 some qualitative properties of minimal solutions of this equation when λ> 0 , p> 0 and to show how the boundary decaying of a works on the behavior of minimal solutions and the pull-in voltage. Particularly, we give a complete analysis for the stability of the minimal solutions.