# 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
##### References:
 [1] J. Appell, E. De Pascale, H.T. Nguyen and P.P. Zabrejko, Multivalued superpositions, Dissertationes Mathematicae, 345 (1995), 1-97. Google Scholar [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. Google Scholar [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. Google Scholar [4] S. Carl and V.K. Le, Existence results for hemivariational inequalities with measures, Applicable Analysis, 86 (2007), 735-753. doi: 10.1080/00036810701397796. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] J. J. Duistermaat and J. A. C. Kolk, Distributions: Theory and Applications Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4675-2. Google Scholar [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. Google Scholar [10] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications Marcel Dekker, Inc. , New York, Basel, Hong Kong, 1995. Google Scholar [11] A. C. Ponce, Selected problems on elliptic equations involving measures, preprint, arXiv:1204.0668v2.Google Scholar [12] M. M. Rao, Measure Theory and Integration Marcel Dekker, Inc. , New York, Basel, 2004. Google Scholar [13] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations American Mathematical Society, Providence, RI, 1997. Google Scholar [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. Google Scholar

show all references

##### References:
 [1] J. Appell, E. De Pascale, H.T. Nguyen and P.P. Zabrejko, Multivalued superpositions, Dissertationes Mathematicae, 345 (1995), 1-97. Google Scholar [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. Google Scholar [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. Google Scholar [4] S. Carl and V.K. Le, Existence results for hemivariational inequalities with measures, Applicable Analysis, 86 (2007), 735-753. doi: 10.1080/00036810701397796. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] J. J. Duistermaat and J. A. C. Kolk, Distributions: Theory and Applications Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4675-2. Google Scholar [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. Google Scholar [10] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications Marcel Dekker, Inc. , New York, Basel, Hong Kong, 1995. Google Scholar [11] A. C. Ponce, Selected problems on elliptic equations involving measures, preprint, arXiv:1204.0668v2.Google Scholar [12] M. M. Rao, Measure Theory and Integration Marcel Dekker, Inc. , New York, Basel, 2004. Google Scholar [13] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations American Mathematical Society, Providence, RI, 1997. Google Scholar [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. Google Scholar
 [1] Ting Li. Pullback attractors for asymptotically upper semicompact non-autonomous multi-valued semiflows. Communications on Pure & Applied Analysis, 2007, 6 (1) : 279-285. doi: 10.3934/cpaa.2007.6.279 [2] Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116 [3] Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061 [4] Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343 [5] Yangrong Li, Renhai Wang, Lianbing She. Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations. Evolution Equations & Control Theory, 2018, 7 (4) : 617-637. doi: 10.3934/eect.2018030 [6] Yejuan Wang, Lin Yang. Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1961-1987. doi: 10.3934/dcdsb.2018257 [7] Yejuan Wang, Tomás Caraballo. Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-24. doi: 10.3934/dcdss.2020092 [8] Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179 [9] Dalila Azzam-Laouir, Warda Belhoula, Charles Castaing, M. D. P. Monteiro Marques. Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators. Evolution Equations & Control Theory, 2020, 9 (1) : 219-254. doi: 10.3934/eect.2020004 [10] Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48 [11] Qi Lü, Xu Zhang. Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application. Mathematical Control & Related Fields, 2018, 8 (1) : 337-381. doi: 10.3934/mcrf.2018014 [12] Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017 [13] Simona Fornaro, Stefano Lisini, Giuseppe Savaré, Giuseppe Toscani. Measure valued solutions of sub-linear diffusion equations with a drift term. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1675-1707. doi: 10.3934/dcds.2012.32.1675 [14] Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020041 [15] Laurent Denis, Anis Matoussi, Jing Zhang. The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5185-5202. doi: 10.3934/dcds.2015.35.5185 [16] Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617 [17] Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911 [18] Huyuan Chen, Feng Zhou. Isolated singularities for elliptic equations with hardy operator and source nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2945-2964. doi: 10.3934/dcds.2018126 [19] Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025 [20] Peter C. Gibson. On the measurement operator for scattering in layered media. Inverse Problems & Imaging, 2017, 11 (1) : 87-97. doi: 10.3934/ipi.2017005

2018 Impact Factor: 0.545

## Metrics

• PDF downloads (22)
• HTML views (110)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]