### 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}

-\mbox{\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.

\begin{equation*}

\begin{cases}

-\mbox{\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.

#### Keywords

Anisotropic Orlicz-Sobolev space; variational inequality; inclusion; multivalued mapping

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