On second order elliptic equations and variational inequalities with anisotropic principal operators

Abstract

This paper is about boundary value problems of the form \begin{equation*} \begin{cases} -{\rm div} [\nabla \Phi(\nabla u)] = f(x,u) &\mbox{in } \Omega, \\ u=0 &\mbox{on } \partial\Omega, \end{cases} \end{equation*} where $\Phi$ is a convex function of $\xi\in \mathbb{R}^N$, rather than a function of the norm $|\xi|$. The problem is formulated appropriately in an anisotropic Orlicz-Sobolev space associated with $\Phi$. We study the existence of solutions and some other properties of the above problem and its corresponding variational inequality in such space.