Regularity of the Extremal Solution of Some Nonlinear Elliptic Problems Involving the p-Laplacian

Abstract

We consider the equation $- {\text{div}}{\left( {{\left| {\nabla u} \right|}^{{p - 2}} \nabla u} \right)} = \lambda f{\left( u \right)}$ on a smooth bounded domain of $\mathbb{R}^{N}$ with zero Dirichlet boundary conditions where p ≥ 2, λ > 0 and f satisfies typical assumptions in the subject of extremal solutions. We prove that, for such general nonlinearities f, the extremal solution u ^* belongs to L ^∞(Ω) if N < p + p/(p − 1) and $u^{*} \in W^{{1,p}}_{0} {\left( \Omega \right)}$ if N < p(1 + p/(p − 1)).