A priori estimates for some elliptic equations involving the $p$-Laplacian

Abstract

We consider the Dirichlet problem for positive solutions of the equation $-{\Delta }_p(u)=f(u)$ in a convex, bounded, smooth domain $\Omega \subset {\mathbb{R}}^N$, with $f$ locally Lipschitz continuous. \par We provide sufficient conditions guarantying $L^{\infty }$ a priori bounds for positive solutions of some elliptic equations involving the $p$-Laplacian and extend the class of known nonlinearities for which the solutions are $L^{\infty }$ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains.