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

2022/07/23 by Huyuan Chen, Ying Wang, Chen, Huyuan +3 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2207.11426

openalex publication_date 2022/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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, N≥ 1, p>0 and a:Ω→[0,1] vanishes at the boundary with the rate \rm dist(x,∂Ω)^γ for γ>0. When p=2 and N=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 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 minimal solutions and the pull-in voltage. Particularly, we give a complete analysis for the stability of the minimal solutions.

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