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

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-Delta_Phi u= g(x,u), & mbox{in}~Omega, u=0, & mbox{on}~partial Omega, end{array} right. end{equation*} where $Delta_{Phi}u :=mbox{div}(phi(| abla u|) abla u)$ and $Omegasubsetmathbb{R}^{N}, N geq 2,$ is a bounded domain with smooth boundary $partialOmega$. 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 Mosers iteration in Orclicz and Orlicz-Sobolev spaces.