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

This paper is concerned with the nonlinear elliptic problem − Δ u = λ ( a − u ) 2in a bounded domain Ω of 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 : Ω ¯ → [ 0 , 1 ]satisfying a ( x ) ≥ κ dist( x , ∂ Ω ) γfor some κ > 0 and γ ∈ ( 0 , 1 ) . Our results show how the boundary decay of the membrane works on the solutions and pull‐in voltage λ.