Abstract
We prove the existence of an entropy solution for a class of nonlinear anisotropic elliptic unilateral problem associated to the following equation
where the right hand side \(\mu \) belongs to \(L^{1}(\Omega )+ W^{-1, \vec {p'}}(\Omega )\). The operator \(-\sum _{i=1}^{N} \partial _i a_i(x,u, \nabla u) \) is a Leray–Lions anisotropic operator and \(\phi _{i} \in C^{0}({\mathbb {R}}, {\mathbb {R}})\).
Similar content being viewed by others
References
Aharouch, L., Akdim, Y.: Strongly nonlinear elliptic unilateral problems without sign condition and \(L^{1}\) data. Appl. Anal. 11–31 (2005)
Akdim, Y., Azroul, E., Benkirane, A.: Existence results for quasilinear degenerated equations via strong convergence of truncations. Rev. Mat. Comlut. 17, 359–379 (2001)
Akdim, Y., Benkirane, A., El Moumni, M.: Existence results for nonlinear elliptic problems with lower order terms. Int. J. Evol. Equ. 8(4), 257–276 (2013)
Akdim, Y., Allalou, C., Salmani, A.: Existence of solutions for some nonlinear elliptic anisotropic unilateral problems with lower order terms. Moroccan J. Pure Appl. Anal. (MJPAA) 4(2), 171–188 (2018). https://doi.org/10.1515/mjpaa-2018-0014
Antontsev, S., Chipot, M.: Anisotropic equations: uniqueness and existence results. Differ. Integral Equ. 6, 401–419 (2008)
Bendahmane, M., Karlsen, K.H.: Anisotropic nonlinear elliptic systems with measure data and anisotropic harmonic maps into spheres. Electron. J. Differ. Equ. 46, 1–30 (2006)
Benkirane, A., Bennouna, J.: Existence of entropy solutions for nonlinear problems in Orlicz spaces. Abstr. Appl. Anal. 7, 85–102 (2002)
Benkirane, A., Bennouna, J.: Existence and uniqueness of solution of unilateral problems with L1-data in Orlicz spaces. Ital. J. Pure Appl. Math. 16, 87–102 (2004)
Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.: An \(L^1\)-theory of existence and uniqueness of nonlinear elliptic equations. Ann. Sc. Norm. Sup. Pisa, CL. IV 22, 240–273 (1995)
Boccardo, L.: Some nonlinear Dirichlet problem in L1 involving lower order terms in divergence form. Progress in elliptic and parabolic partial differential equations (Capri, 1994), Pitman Res. Notes Math. Ser., 350, pp. 43–57, Longman, Harlow (1996)
Boccardo, L., Gallouët, T.: Strongly nonlinear elliptic equations having natural growth terms and \(L^{1}\) data. Nonlinear Anal. T.M.A. 19, 573–578 (1992)
Boccardo, L., Gallouët, T., Marcellini, P.: Anisotropic equations in \(L^{1}\). Differ. Integral Equ. 1, 209–212 (1996)
Boccardo, L., Gallouët, T., Orsina, L.: Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Anal. Math. 73, 203–223 (1997)
Boccardo, L., Marcellini, P., Sbordone, C.: \(L ^{\infty }\)-regularity for variational problems with sharp non standard growth conditions. Boll. Un. Mat. It. Sez. A(7), 4-A (1990)
Boccardo, L., Murat, F., Puel, J.P.: Existence of bounded solution for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. 152, 183–196 (1988)
Cupini, G., Marcellini, P., Mascolo, E.: Regularity of minimizers under limit growth conditions. Nonlinear Anal. (2017). https://doi.org/10.1016/j.na.2016.06.002
Di Castro, A.: Existence and regularity results for anisotropic elliptic problems. Adv. Nonlinear Stud. 9, 367–393 (2009)
Di Castro, A.: Anisotropic elliptic problems wih natural growth terms. Manuscripta Math. 135, 521–543 (2011)
Di Nardo, R., Feo, F.: Existence and uniqueness for nonlinear anisotropic elliptic equations. Arch. Math. 102, 141–153 (2014)
Fragala, I., Gazzola, F., Kawohl, B.: Existence and nonexistence results for anisotropic quasilinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 5, 715–734 (2004)
Guibé, O., Mercaldo, A.: Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Commun. Pure Appl. Anal. 1, 163–192 (2008)
Leray, J., Lions, L.: Quelques Méthodes de Résolution des Problémes aux Limites Non linéaires. Dunod, Paris (1968)
Li, F.Q.: Anisotropic elliptic equations in \(L^{m}\). J. Convex Anal. 2, 417–422 (2001)
Troisi, M.: Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche Mat. 18, 3–24 (1969)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, salmani abdelhafid as the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Salmani, A., Akdim, Y. & Redwane, H. Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data. Ricerche mat 69, 121–151 (2020). https://doi.org/10.1007/s11587-019-00452-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-019-00452-0