Regularity of stable radial solutions to semilinear elliptic equations in MEMS problems

Abstract

This paper investigates the regularity of stable radial solutions to semilinear elliptic equations arising in MEMS problems, modeled by the Dirichlet problem $-\Delta u=f(u)$ in the unit ball $B_1$, where the nonlinearity $f\in C^1([0,1))$ is nonnegative and satisfies $\int^1_0f(s)\,ds=+\infty$. We focus on the case where $f$ blows up as $u\to 1^{-}$. Micro-electro-mechanical systems (MEMS) are widely used devices in engineering and technology. Our main result establishes for dimensions $2\le n\le 6$, every stable radial solution is regular, meaning $\|u\|_{L^{\infty}(B_1)}<1$. This result gives a positive answer to an open problem posed by Bruera and Cabr\'e concerning the regularity of stable solutions for singular nonlinearities without requiring a Crandall-Rabinowitz type condition, at least in the radial case.