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

作者：Chen, Huyuan^*; Wang, Ying; Zhou, Feng

来源：Zeitschrift für Angewandte Mathematik und Physik, 2024, 75(1): 3.

DOI：10.1007/s00033-023-02116-4

• MEMS equation
• Stability
• Pull-in voltage

This paper is concerned with the elliptic equation -Delta u=lambda/(a-u)p in a connected, bounded C-2 domain Omega of R-N subject to zero Dirichlet boundary conditions, where lambda>0, p>0 and a:Omega<overline>->[0,1] vanishes at the boundary with the rate dist(x,partial derivative Omega)(gamma) for gamma>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 lambda>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.