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

TL;DR

This paper establishes conditions under which positive solutions to certain p-Laplacian elliptic equations in convex domains are bounded, extending known results and ensuring existence of solutions.

Contribution

It provides new sufficient conditions for a priori bounds and extends the class of nonlinearities for which solutions are bounded, leading to existence results.

Findings

Established $L^{ abla} $ bounds for solutions

Extended class of nonlinearities with bounded solutions

Proved existence of positive solutions in convex domains

Abstract

We consider the Dirichlet problem for positive solutions of the equation $-\Delta_p (u) = f(u)$ in a convex, bounded, smooth domain $\Omega \subset\R^N$, 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.