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

Abstract

This paper is concerned with the nonlinear elliptic problem −Δu=λfalse(a−ufalse)2 in a bounded domain Ω of RN 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:normalΩ¯→false[0,1false] satisfying afalse(xfalse)≥κ dist (x,∂Ω)γ for 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.

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

“…The difficulty to obtain the stability of u p,λ comes from the blowing up of 1 (a−u p,λ ) p+1 at the boundary. [4] obtains the stability for only λ ∈ (0, λ * ), where λ * ≤ λ * γ,p and only getting the equal for the particular case γ = 2 3 . From Theorem 1.4 we give a complete stability for all minimal solutions in a general model.…”

“…It is proved in [4] that for γ ∈ (0, 2 3 ] there exists a critical value λ * (pull-in voltage) depending on κ, γ such that if λ ∈ (0, λ * ), problem (1.3) has a minimal solution, while for λ > λ * , no solution exists for (1.3).…”