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

This paper is concerned with the nonlinear elliptic problem −Δ𝑢 = 𝜆/(𝑎−𝑢)2 in 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 ∶ Ω̄ → [0, 1] satisfying 𝑎(𝑥) ≥ 𝜅dist(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.