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

The objective of our paper is to investigate fractional elliptic equations of the form $(-\mathrm{\Delta }{)}^{s}u=\frac{\lambda }{(a-u{)}^2}$ within a bounded domain $\mathrm{\Omega }$, 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.