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

$$\begin{aligned} \left\{ \ \begin{array}{ll} \displaystyle -\Delta _\Phi u= g(x,u), &{} \hbox {in}~\Omega ,\\ u=0, &{} \hbox {on}~\partial \Omega , \end{array} \right. \end{aligned}$$

where $\Delta _{\Phi }u :=\hbox {div}(\phi (|\nabla u|)\nabla u)$ 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 Orlicz and Orlicz-Sobolev spaces.