Revised regularity results for quasilinear elliptic problems driven by the $\Phi$-Laplacian operator

Abstract

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $\Phi$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-Δ_Φu= g(x,u), & \mbox{in}~Ω, u=0, & \mbox{on}~\partial Ω, \end{array} \right. \end{equation*} where $Δ_Φu :=\mbox{div}(ϕ(|\nabla u|)\nabla u)$ and $\Omega \subset {\mathbb{R}}^N,N\ge 2,$ is a bounded domain with smooth boundary $\partial \Omega$. Our work concerns on nonlinearities $g$ which can be homogeneous or non-homogeneous. For the homogeneous case we consider an existence result together with a regularity result proving that any weak solution remains bounded. Furthermore, for the non-homogeneous case, the nonlinear term $g$ can be subcritical or critical proving also that any weak solution is bounded. The proofs are based on Moser's iteration in Orclicz and Orlicz-Sobolev spaces.

Here we consider some regularity results for quasilinear elliptic problems involving nonhomoegeneous operators