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

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.