Existence and nonexistence of solutions for fractional elliptic equations arising from closed MEMS model

TL;DR

This paper studies fractional elliptic equations modeling MEMS devices, establishing conditions for the existence or nonexistence of solutions based on boundary behavior and parameters, thus advancing understanding of MEMS device stability.

Contribution

It provides new existence and nonexistence criteria for fractional MEMS equations considering boundary decay effects and parameter influences.

Findings

Existence of solutions depends on the parameter nd boundary decay conditions.

Nonexistence results are established for certain parameter ranges.

The boundary behavior of the membrane significantly affects solution existence.

Abstract

The objective of our paper is to investigate fractional elliptic equations of the form $(-\Delta)^s u=\frac{\lambda }{(a-u)^2}$ within a bounded domain $\Omega$, subject to zero Dirichlet boundary conditions. Here, $s\in(0,1)$, $\lambda>0$, and the function $a$ vanishes at the boundary while satisfying additional conditions. This problem originates from Micro-Electromechanical Systems (MEMS) devices, particularly when the elastic membrane makes contact with the ground plate at the boundary. We establish both existence and nonexistence results, illustrating how the boundary decay of the membrane influences the solutions and pull-in voltage.