Abstract
In this paper we prove existence results for distributional solutions of nonlinear elliptic systems with a measure data. The functional setting involves Lebesgue-Sobolev spaces as well as weak Lebesgue (Marcinkiewicz) spaces with variable exponents \(W_0^{1,p( \cdot )}(\Omega )\) and \({M^{p( \cdot )}}(\Omega )\)respectively.
Article PDF
Similar content being viewed by others
References
E. Acerbi and G. Mingione. Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal, 156:121–140, 2001.
E. Azroul, M. B. Benboubker and M. Rhoudaf. On some p(x)-quasilinear problem with right-hand side measure. Math. Comput. Simul, (102):117-130, 2014.
M. Bendahmane and K. Karlsen. Anisotropic nonlinear elliptic systems with measure data and anisotropic harmonic maps into spheres. Electronic Journal of Differential Equations, 1-29, 2006.
M. Bendahmane, P. Wittbold. Renormalized solutions for nonlinear elliptic equations with variable exponents and L 1-data. Nonlinear Analysis TMA 70(2):567–583, 2009.
P. Benilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, and J. L. Vazquez. An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22(2):241273, 1995.
Y. Chen, Slevine, and M. Rao., Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math., 66:1383–1406, 2006.
L. Diening. Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces L p (·) and W k;p (·). Mathematische Nachrichten, 268(1):31–43, 2004.
G. Dolzmann, N. Hungerbühler, and S. Müller. Non-linear elliptic systems with measure-valued right hand side. Math. Z, 226(4):545–574, 1997.
X.L. Fan and D. Zhao. On the spaces L p (x) (U) and W m;p (x)(U). Math. Anal. Appl, (263):424-446, 2001.
R. Landes. Test functions for elliptic systems and maximum principles. Forum Math, 12(1):23–52, 2000.
F. Mokhtari, K. Bachouche, and H. Abdelaziz Nonlinear elliptic equations with variable exponents and measure or L m data, Journal of Mathematical Sciences: Advances and Applications 35:73–101, 2015.
K. Rajagopal, M. Ružička. Mathematical modelling of electro-rheological fluids. Contin. Mech. Thermodyn, 13:59–78, 2001.
M. Ružička. Electrorheological fluids:modeling and mathematical theory, Springer, Berlin. Lecture Notes in Mathematics, 1748, 2000.
M. Sanchóon and J.M. Urbano. Entropy solutions for the p(x)-Laplace equation. Trans. Amer. Math. Soc. 361:6387–6405, 2009.
C. Zhang and S. Zhou, Entropy and renormalized solutions for the p(x)-Laplacian equation with measure data, Bull. Aust. Math. Soc. 82:459–479, 2010.
C. Zhang, Entropy solutions for nonlinear elliptic equations with variable exponents, Electronic Journal of Differential Equations. 92:1–14, 2014.
V.V. Zhikov. On the density of smooth functions in Sobolev-Orlicz spaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Math. Inst. Steklov. (POMI), 310:67-81, 226, 2004.
Acknowledgment
The authors would like to thank the referee for his comments and suggestions. M. Bendahmane is supported by the Moroccan project FINCOME.
Author information
Authors and Affiliations
Corresponding authors
Additional information
2010 Mathematics Subject Classification. 35K05,35K55.
a Institut de Mathématiques de Bordeaux UMR CNRS 5251, Université Victor Segalen Bordeaux 2, F-33076 Bordeaux Cedex, France.
e-mail: mostafa.bendahmane@u-bordeaux2.fr
b Laboratoire des EDPNL, ENS-Kouba, Department of Mathematics and Informatics. Faculty of Sciences, Univ of Benyoucef Benkhadda, 2 rue Didouche Mourad, Algiers, Algeria.
e-mail: fares_maths@yahoo.fr
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bendahmane, M., Mokhtari, F. Nonlinear elliptic systems with variable exponents and measure data. Moroc J Pure Appl Anal 1, 8 (2015). https://doi.org/10.7603/s40956-015-0008-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.7603/s40956-015-0008-3