On quasi-variational inequalities with discontinuous multivalued lower order terms given by bifunctions

Abstract

In this paper, we consider the existence and some qualitative properties of solutions to quasi-variational inequalities of the form $$ \begin{cases} \langle {\mathcal A}(u) , v-u \rangle + \langle F(u,u) , v-u\rangle + J(v,u) - J(u,u) \ge 0,\;\forall v\in W^{1,p}(\Omega) , \\ u\in D(J(\cdot, u)) , \end{cases} $$ where $\Omega$ is a bounded domain in ${\mathbb R}^N$, ${\mathcal A}$ is a second-order elliptic operator of Leray--Lions type on $W^{1,p}(\Omega)$, $F(\cdot , \cdot)$ is a multivalued bifunction, and $J(\cdot , u)$ is a convex functional depending on $u$. We propose concepts of sub- and supersolutions for this problem and study the existence and enclosure of solutions, and also the existence of extremal solutions between its sub- and supersolutions. Properties and examples of the involved mappings $F$ and $J$ and of sub-supersolutions of the above quasi-variational inequality are also presented.