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

TL;DR

This paper proves that all stable radial solutions to certain singular semilinear elliptic equations in dimensions 2 to 6 are regular, addressing an open problem in the mathematical analysis of MEMS models.

Contribution

It establishes the regularity of stable radial solutions for a class of singular nonlinearities in MEMS problems without additional conditions, in dimensions 2 to 6.

Findings

Stable solutions are regular in dimensions 2 to 6.

Addresses an open problem by Bruera and Cabré.

Does not require Crandall-Rabinowitz condition.

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.