Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data


We prove the existence of an entropy solution for a class of nonlinear anisotropic elliptic unilateral problem associated to the following equation

$$\begin{aligned} -\sum _{i=1}^{N} \partial _i a_i(x,u, \nabla u) -\sum _{i=1}^{N}\partial _{i}\phi _{i}( u)=\mu , \end{aligned}$$

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 \(\phi _{i} \in C^{0}({\mathbb {R}}, {\mathbb {R}})\).

This is a preview of subscription content, log in to check access.


  1. 1.

    Aharouch, L., Akdim, Y.: Strongly nonlinear elliptic unilateral problems without sign condition and \(L^{1}\) data. Appl. Anal. 11–31 (2005)

  2. 2.

    Akdim, Y., Azroul, E., Benkirane, A.: Existence results for quasilinear degenerated equations via strong convergence of truncations. Rev. Mat. Comlut. 17, 359–379 (2001)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    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)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    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

    Article  Google Scholar 

  5. 5.

    Antontsev, S., Chipot, M.: Anisotropic equations: uniqueness and existence results. Differ. Integral Equ. 6, 401–419 (2008)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    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)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Benkirane, A., Bennouna, J.: Existence of entropy solutions for nonlinear problems in Orlicz spaces. Abstr. Appl. Anal. 7, 85–102 (2002)

    MathSciNet  Article  Google Scholar 

  8. 8.

    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)

    MATH  Google Scholar 

  9. 9.

    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)

    MATH  Google Scholar 

  10. 10.

    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)

  11. 11.

    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)

    Article  Google Scholar 

  12. 12.

    Boccardo, L., Gallouët, T., Marcellini, P.: Anisotropic equations in \(L^{1}\). Differ. Integral Equ. 1, 209–212 (1996)

    MATH  Google Scholar 

  13. 13.

    Boccardo, L., Gallouët, T., Orsina, L.: Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Anal. Math. 73, 203–223 (1997)

    MathSciNet  Article  Google Scholar 

  14. 14.

    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)

    MATH  Google Scholar 

  15. 15.

    Boccardo, L., Murat, F., Puel, J.P.: Existence of bounded solution for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. 152, 183–196 (1988)

    MathSciNet  Article  Google Scholar 

  16. 16.

    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

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Di Castro, A.: Existence and regularity results for anisotropic elliptic problems. Adv. Nonlinear Stud. 9, 367–393 (2009)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Di Castro, A.: Anisotropic elliptic problems wih natural growth terms. Manuscripta Math. 135, 521–543 (2011)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Di Nardo, R., Feo, F.: Existence and uniqueness for nonlinear anisotropic elliptic equations. Arch. Math. 102, 141–153 (2014)

    MathSciNet  Article  Google Scholar 

  20. 20.

    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)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Guibé, O., Mercaldo, A.: Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms. Commun. Pure Appl. Anal. 1, 163–192 (2008)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Leray, J., Lions, L.: Quelques Méthodes de Résolution des Problémes aux Limites Non linéaires. Dunod, Paris (1968)

    Google Scholar 

  23. 23.

    Li, F.Q.: Anisotropic elliptic equations in \(L^{m}\). J. Convex Anal. 2, 417–422 (2001)

    MATH  Google Scholar 

  24. 24.

    Troisi, M.: Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche Mat. 18, 3–24 (1969)

    MathSciNet  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Abdelhafid Salmani.

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

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation


  • Entropy solutions
  • Anisotropic elliptic equations
  • Anisotropic Sobolev space

Mathematics Subject Classification

  • 35J60
  • 35J87
  • 35J66