# American Institute of Mathematical Sciences

April  2018, 11(2): 193-212. doi: 10.3934/dcdss.2018012

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

 Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle, Germany

* Corresponding author: Christoph Tietz

Received  November 2016 Revised  April 2017 Published  January 2018

Fund Project: The second author is supported by a doctoral studies grant of Saxony-Anhalt.

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.
Citation: Siegfried Carl, Christoph Tietz. Quasilinear elliptic equations with measures and multi-valued lower order terms. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 193-212. doi: 10.3934/dcdss.2018012
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