SpringerLink
Log in

Renormalized Solution to Nonlinear Elliptic Equations with Measure Data in Musielak Spaces

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

This article is concerned with the existence of renormalized solution for an elliptic equation having two lower order terms and measure data in Musielak–Orlicz spaces.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Aharouch, L., Bennouna, J., Touzani, A.: Existence of Renormalized Solution of Some Elliptic Problems in Orlicz Spaces. Rev. Mat. Complut. 22(1), 91–110 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aissaoui Fqayeh, A., Benkirane, A., El Moumni, M., Youssfi, A.: Existence of renormalized solutions for some strongly nonlinear elliptic equations in Orlicz spaces. Geor. Math. J. 22(3), 305–321 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ait Khellou, M., Benkirane, A.: Renormalized solution for nonlinear elliptic problems with lower order terms and \(L^1\) data in Musielak–Orlicz spaces. Ann. Univ. Craiova Math. Comput. Sci. Ser. 43(2), 164–187 (2016)

    MATH  Google Scholar 

  5. Ait Khellou, M., Benkirane, A., Douiri, S.. M.: Strongly non-linear elliptic problems in Musielak spaces with \(L^1\) data. Nonlinear Stud. 23(3), 491–510 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Ait Khellou, M., Benkirane, A.., Douiri, S.. M.: Some main properties of Musielak spaces with only the log-Hölder continuity condition. Ann. Funct. Anal. 11, 1062–1080 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Azroul, E., Barbara, A., Benboubker, M.B., Ouaro, S.: Renormalized solutions for a \(p(x)\)-Laplacian equation with Neumann nonhomogeneous boundary conditions and \(L^1\)-data. Ann. Univ. Craiova Math. Comput. Sci. Ser. 40(1), 9–22 (2013)

    MATH  Google Scholar 

  8. Bendahmane, M., Wittbold, P.: Renormalized solutions for nonlinear elliptic equations with variable exponents and \(L^1\) data. Nonlinear Anal. 70, 567–583 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Benkirane, A., Bennouna, J.: Existence of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms in Orlicz spaces, Partial differential equations. Lecture Notes in Pure and Appl. Math., Dekker, New York 229, 125–138 (2002)

  10. Benkirane, A., Elmahi, A.: Almost everywhere convergence of gradients of solutions to elliptic equations in Orlicz spaces and application. Nonlinear Anal. Theory Methods Appl. 28, 1769–1784 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Benkirane, A., Sidi El Vally, M.: (Ould Mohamedhen Val): Variational inequalities in Musielak–Orlicz–Sobolev spaces. Bull. Belg. Math. Soc. Simon Stevin 21(5), 787–811 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Boccardo, L., Giachetti, D., Diaz, J.I., Murat, F.: Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. J. Differ. Equ. 106(2), 215–237 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bourahma, M., Benkirane, A., Bennouna, J.: A strongly nonlinear elliptic problem with generalized growth in Musielak spaces. Differ. Equ. Dyn. Syst. (2021). https://doi.org/10.1007/s12591-020-00558-0

    Article  MATH  Google Scholar 

  14. Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28(4), 741–808 (1999)

    MathSciNet  MATH  Google Scholar 

  15. Diening, L., Harjulehto, P., Hästö, P., Råžička, M.: Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics (Vol.2017), Springer, Heidelberg, Germany, (2011)

  16. DiPerna, R.J., Lions, P.L.: On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. (2) 130(2), 321–366 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Elarabi, R., Rhoudaf, M., Sabiki, H.: Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces. Ricerche Mat. 67, 549–579 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gossez, J.P.: Some approximation properties in Orlicz-Sobolev spaces. Stud. Math. 74(1), 17–24 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gwiazda, P., Swierczewska-Gwiazda, A.: On non-Newtonian fluids with a property of rapid thickening under different stimulus. Math. Models Methods Appl. Sci. 18(7), 1073–1092 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gwiazda, P., Wittbold, P., Wróblewska, A., Zimmermann, A.: Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces. J. Differ. Equ. 253(2), 635–666 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mabdaoui, M., Moussa, H., Rhoudaf, M.: Entropy solutions for a nonlinear parabolic problems with lower order term in Orlicz spaces. Anal. Math. Phys. 7, 47–76 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Malik, J., Perona, P.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)

    Article  Google Scholar 

  23. Musielak, J.: Modular spaces and Orlicz spaces , Lecture Notes in Math. 1034, (1983)

  24. Rajagopal, K.R., Ružička, M.: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn. 13, 59–78 (2001)

    Article  MATH  Google Scholar 

  25. Rakotoson, J.M.: Uniqueness of renormalized solutions in a T-set for the \(L^1\)-data problem and the link between various formulations. Indiana Univ. Math. J. 43(2), 685–702 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ružička, M.: Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics. Springer, Berlin (2000)

Download references

Author information

Authors and Affiliations

  1. Department of Sciences, Higher Normal School, Moulay Ismaïl University of Meknes, P.O. Box 3104, Toulal, Meknes, 50000, Morocco

    Mustafa Ait Khellou

Authors
  1. Mustafa Ait Khellou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mustafa Ait Khellou.

Ethics declarations

Conflict of interest

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

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ait Khellou, M. Renormalized Solution to Nonlinear Elliptic Equations with Measure Data in Musielak Spaces. Differ Equ Dyn Syst (2023). https://doi.org/10.1007/s12591-023-00655-w

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12591-023-00655-w

Keywords

Mathematics Subject Classification

Search

Navigation