About positive $W_{\rm loc}^{1,\Phi}(\Omega)$-solutions to quasilinear elliptic problems with singular semilinear term

Abstract

This paper deals with the existence, uniqueness and regularity of positive $W_{\rm loc}^{1,\Phi}(\Omega)$-solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $\Phi$-Laplacian operator. The proof of the existence is based on a variant of the generalized Galerkin method that we developed inspired by ideas of Browder [4] and a comparison principle. By the use of a kind of Moser's iteration scheme we show the $L^{\infty}(\Omega)$-regularity for positive solutions.