# 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 and Continuous Dynamical Systems - S, 2018, 11 (2) : 193-212. doi: 10.3934/dcdss.2018012
##### References:
 [1] J. Appell, E. De Pascale, H.T. Nguyen and P.P. Zabrejko, Multivalued superpositions, Dissertationes Mathematicae, 345 (1995), 1-97. [2] L. Boccardo, Some nonlinear Dirichlet problems in L1 involving lower order terms in divergence form, Progress in Elliptic and Parabolic Partial Differential Equations (Capri, 1994), 43–57, Pitman Res. Notes Math. Ser. , 350, Longman, Harlow, 1996. [3] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, Journal of Functional Analysis, 87 (1989), 149-169.  doi: 10.1016/0022-1236(89)90005-0. [4] S. Carl and V.K. Le, Existence results for hemivariational inequalities with measures, Applicable Analysis, 86 (2007), 735-753.  doi: 10.1080/00036810701397796. [5] S. Carl and V.K. Le, Elliptic inequalities with multi-valued operators: Existence, comparison and related variational-hemivariational type inequalities, Nonlinear Analysis: Theory, Methods & Applications, 121 (2015), 130-152.  doi: 10.1016/j.na.2014.10.033. [6] S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities Springer Monograph in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3. [7] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, nnali della Scuola Normale Superiore di Pisa -Classe di Scienze, 28 (1999), 741-808. [8] J. J. Duistermaat and J. A. C. Kolk, Distributions: Theory and Applications Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4675-2. [9] J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder, Bulletin de la Société Mathématique de France, 93 (1965), 97-107. [10] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications Marcel Dekker, Inc. , New York, Basel, Hong Kong, 1995. [11] A. C. Ponce, Selected problems on elliptic equations involving measures, preprint, arXiv:1204.0668v2. [12] M. M. Rao, Measure Theory and Integration Marcel Dekker, Inc. , New York, Basel, 2004. [13] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations American Mathematical Society, Providence, RI, 1997. [14] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions Pitman Monographs and Surveys in Pure and Applied Mathematikcs, 75 2nd edition, Longman, New York, 1995.

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##### References:
 [1] J. Appell, E. De Pascale, H.T. Nguyen and P.P. Zabrejko, Multivalued superpositions, Dissertationes Mathematicae, 345 (1995), 1-97. [2] L. Boccardo, Some nonlinear Dirichlet problems in L1 involving lower order terms in divergence form, Progress in Elliptic and Parabolic Partial Differential Equations (Capri, 1994), 43–57, Pitman Res. Notes Math. Ser. , 350, Longman, Harlow, 1996. [3] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, Journal of Functional Analysis, 87 (1989), 149-169.  doi: 10.1016/0022-1236(89)90005-0. [4] S. Carl and V.K. Le, Existence results for hemivariational inequalities with measures, Applicable Analysis, 86 (2007), 735-753.  doi: 10.1080/00036810701397796. [5] S. Carl and V.K. Le, Elliptic inequalities with multi-valued operators: Existence, comparison and related variational-hemivariational type inequalities, Nonlinear Analysis: Theory, Methods & Applications, 121 (2015), 130-152.  doi: 10.1016/j.na.2014.10.033. [6] S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities Springer Monograph in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3. [7] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, nnali della Scuola Normale Superiore di Pisa -Classe di Scienze, 28 (1999), 741-808. [8] J. J. Duistermaat and J. A. C. Kolk, Distributions: Theory and Applications Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4675-2. [9] J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder, Bulletin de la Société Mathématique de France, 93 (1965), 97-107. [10] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications Marcel Dekker, Inc. , New York, Basel, Hong Kong, 1995. [11] A. C. Ponce, Selected problems on elliptic equations involving measures, preprint, arXiv:1204.0668v2. [12] M. M. Rao, Measure Theory and Integration Marcel Dekker, Inc. , New York, Basel, 2004. [13] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations American Mathematical Society, Providence, RI, 1997. [14] I. I. Vrabie, Compactness Methods for Nonlinear Evolutions Pitman Monographs and Surveys in Pure and Applied Mathematikcs, 75 2nd edition, Longman, New York, 1995.
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