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

Carlos Alberto Santos, José Valdo Gonçalves, Marcos Leandro Carvalho

DOI: http://dx.doi.org/10.12775/TMNA.2019.009


This paper deals with the existence, uniqueness and regularity of positive
$W_{\rom loc}^{1,\Phi}(\Omega)$-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 \cite{Browder} 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.


$\Phi$-Laplacian; Orlicz--Sobolev spaces; singular, Galerkin method

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