On the smoothness of weak solutions to subcritical semilinear elliptic equations in any dimension

TL;DR

This paper establishes conditions under which semi-stable weak solutions to subcritical semilinear elliptic equations are smooth and classical in any dimension, broadening the class of nonlinearities for which regularity can be guaranteed.

Contribution

It provides new sufficient conditions ensuring regularity of semi-stable solutions for a wider class of subcritical nonlinearities in any dimension.

Findings

Semi-stable solutions are smooth under the new conditions.

Classical solutions are obtained from weak solutions.

Broader class of nonlinearities included, beyond power functions.

Abstract

Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions guarantying that semi-stable weak positive solutions to subcritical semilinear elliptic equations are smooth in any dimension, and as a consequence, classical solutions. By a subcritical nonlinearity we mean $f(x,s)/s^\frac{N+2}{N-2} \to 0$ as $s\to\infty$, including non-power nonlinearities, and enlarging the class of subcritical nonlinearities, which is usually reserved for power like nonlinearities.