Quasilinear elliptic equations with measures and multi-valued lower order terms

Abstract

In this paper, we consider the existence and further qualitative properties of solutions of the Dirichlet problem to quasilinear multi-valued elliptic equations with measures of the form$Au + G(\cdot,u) \ni f,$where $A$ is a second order elliptic operator of Leray-Lions type and $f\in \mathcal M_b(\Omega)$ is a given Radon measure on a bounded domain $\Omega\subset \mathbb R^N$. The lower order term $s\mapsto G(\cdot,s)$ is assumed to be a multi-valued upper semicontinuous function, which includes Clarke's gradient $s\mapsto \partial j(\cdot,s)$ of some locally Lipschitz function $s\mapsto j(\cdot,s)$ as a special case. Our main goals and the novelties of this paper are as follows: First, we develop an existence theory for the above multi-valued elliptic problem with measure right-hand side. Second, we propose concepts of sub-supersolutions for this problem and establish an existence and comparison principle. Third, we topologically characterize the solution set enclosed by sub-supersolutions.