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

Abstract

Let us consider a semilinear boundary value problem −∆u =f(x, u), in Ω, with Dirichlet boundary conditions, where Ω ⊂ 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 N+2 N−2 → 0 as s → ∞, including non-power nonlinearities, and enlarging the class of subcritical nonlinearities, which is usually reserved for power like nonlinearities.