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

Siegfried Carl; Christoph Tietz

doi:10.3934/dcdss.2018012

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2018, Volume 11, Issue 2: 193-212. Doi: 10.3934/dcdss.2018012

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

Siegfried Carl and
Christoph Tietz^,

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

* Corresponding author: Christoph Tietz
* Corresponding author: Christoph Tietz

Received: October 31, 2016

Revised: March 31, 2017

Published: April 2018

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

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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.

Keywords:

Upper semicontinuous multi-valued operator,
pseudomonotone multi-valued operator,
Radon measure,
sub-supersolutions,
quasilinear elliptic operator.

Mathematics Subject Classification: Primary: 35R70, 35R06; Secondary: 35J62, 47H05.

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