Existence and regularity results for nonlinear elliptic equations in Orlicz spaces

We are concerned with the existence and regularity of the solutions to the Dirichlet problem, for a class of quasilinear elliptic equations driven by a general differential operator, depending on (x,u, backward difference u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,u,\nabla u)$$\end{document}, and with a convective term f. The assumptions on the members of the equation are formulated in terms of Young's functions, therefore we work in the Orlicz-Sobolev spaces. After establishing some auxiliary properties, that seem new in our context, we present two existence and two regularity results. We conclude with several examples.