Existence and regularity of positive solutions of quasilinear elliptic problems with singular semilinear term

TL;DR

This paper establishes the existence and regularity of positive solutions for singular quasilinear elliptic problems involving the -Laplacian operator, using a generalized Galerkin method and Moser iteration.

Contribution

It introduces a novel approach combining a generalized Galerkin method and Moser iteration to prove existence and regularity for singular elliptic problems with the -Laplacian.

Findings

Existence of positive solutions under specified conditions

Solutions are bounded in -infinity norm

Method can be applied to similar singular elliptic problems

Abstract

This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $\Phi$-Laplacian operator. The proof of existence is based on a variant of the generalized Galerkin method that we developed inspired on ideas by Browder and a comparison principle. By using a kind of Moser iteration scheme we show $L^{\infty}(\Omega)$-regularity for positive solutions