Skip to main content
SpringerLink
Log in

Equivalence of Entropy and Renormalized Solutions of Anisotropic Elliptic Problem in Unbounded Domains with Measure Data

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We consider a class of anisotropic elliptic equations of second order with variable exponents of non-linearity where a special Radon measure is used as the right-hand side. We establish uniqueness of entropy and renormalized solutions of the Dirichlet problem in anisotropic Sobolev spaces with variable exponents of non-linearity for arbitrary domains and certain other their properties. In addition, we prove the equivalence of entropy and renormalized solutions of the problem under consideration.

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

Access this article

Log in via an institution

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

    MathSciNet  MATH  Google Scholar 

  2. Malusa, A. “A new proof of the stability of renormalized solutions to elliptic equations with measure data”, Asymptotic Anal. 43 (1-2), 111–129 (2005).

    MathSciNet  MATH  Google Scholar 

  3. Bidaut-Verón, M.F. “Removable singularities and existence for a quasilinear equation with absorption or source term and measure data”, Adv. Nonlinear Stud. 3 (1), 25–63 (2003).

    Article  MathSciNet  Google Scholar 

  4. Malusa, A., Porzio, M.M. “Renormalized solutions to elliptic equations with measure data in unbounded domains”, Nonlinear Anal. 67 (8), 2370–2389 (2007).

    Google Scholar 

  5. Zhikov, V.V. “Variation problems and non-lineary elliptic equations with non-standard growth condi¬tions”, Probl. mathem. analys. 54, 23–112 (2011).

    Google Scholar 

  6. Alhutov, Yu.A. “Hölder continuity of p(x)-harmonic functions”, Sb. Math. 196 (2), 147–171 (2005).

    Article  MathSciNet  Google Scholar 

  7. Diening, L., Harjulehto, P., Hästö, P.M., Růžička, M. Lebesgue and Sobolev spaces with variable exponents (Springer, New York, 2011).

    Book  Google Scholar 

  8. Nyanquini, I., Ouaro, S., Soma, S. “Entropy solution to nonlinear multivalued elliptic problem with variable exponents and measure data”, Annals Univ. Craiova, Math, and Comp. Sci. Ser. 40 (2), 1–25 (2013).

    MathSciNet  MATH  Google Scholar 

  9. Zhang, C. “Entropy solutions to nonlinear elliptic equations with variable exponents”, Electron. J. Diff. Equat. 2014 (92), 1–14 (2014).

    MathSciNet  Google Scholar 

  10. Benboubker, M.B., Chrayteh, H., El Moumni, M., Hjiaj, H. “Entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data”, Acta Math. Sinica, English Ser. 31 (1), 151–169 (2015).

    Article  MathSciNet  Google Scholar 

  11. Ahmedatt, T., Azroul, E., Hjiaj, H., Touzani, A. “Existence of Entropy solutions for some nonlinear elliptic problems involving variable exponent and measure data”, Bol. Soc. Paran. Mat. 36 (2), 33–55 (2018).

    Article  MathSciNet  Google Scholar 

  12. Zhang, C., Zhou, S. “Entropy and renormalized solutions for the p(x)-laplacian equation with measure data”, Bull. Aust. Math. Soc. 82 (3), 459–479 (2010).

    Article  MathSciNet  Google Scholar 

  13. Konaté, I., Ouaro, S. “Good Radon measure for anisotropic problems with variable exponent”, Electron. J. Diff. Equat. 2016 (221), 1–19 (2016).

    MathSciNet  MATH  Google Scholar 

  14. Bénilan, Ph., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.L. “An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations”, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 22 (2), 241–273 (1995).

    MATH  Google Scholar 

  15. Kozhevnikova, L.M. “Entropy solution of elliptic problem in anisotropic Sobolev-Orlitz spaces”, J. vychisl. matem i mat. phys. 57 (3), 429–447 (2017).

    Google Scholar 

  16. Kozhevnikova, L.M. “Existence of entropy solutions of elliptic problem in anisotropic Sobolev-Orlitz spaces”, Itogi nauki i techn. Ser. Sovr. matem. i ee prilozh. Temat. obzor. 139, 15–38 (2017).

    Google Scholar 

  17. Kozhevnikova, L.M. “Entropy solutions of anisotropic elliptic equations with variable exponents of non-linearities in unbounded domains”, Sovr. matem. Fundamental, napravleniya. 63 (3), 475–493 (2017).

    Google Scholar 

  18. Kozhevnikova, L.M. “Entropy and renormalized solutions of anisotropic elliptic equations with variable nonlinearity exponents”, Sb. Math. 210 (3), 417–446 (2019).

    Article  MathSciNet  Google Scholar 

  19. Mukminov, F.H. “Uniqueness of the renormalized solution of an elliptic-parabolic problem in anisotropic Sobolev-Orlicz spaces”, Sb. Math. 208 (8), 1187–1206 (2017).

    Article  MathSciNet  Google Scholar 

  20. Fan, X. “Anisotropic variable exponent Sobolev spaces and p(x)-Laplacian equations”, Complex Var. Elliptic Equat. 56 (7-9), 623–642 (2011).

    Article  MathSciNet  Google Scholar 

  21. Benboubker, M.B., Azroul, E., Barbara, A. “Quasilinear elliptic problems with nonstandard growths”, Electron. J. Diff. Equat. 2011 (62), 1–16 (2011).

    MathSciNet  MATH  Google Scholar 

Download references

Funding

The research is performed with support of Russian Foundation for Basic Researches (grant no. 18-01-00428).

Author information

Authors and Affiliations

  1. Sterlitamak Branch of Bashkir State University, 31 Lenin Ave., Sterlitamak, 453103, Russia

    L. M. Kozhevnikova

  2. Elabuga Branch of Kazan Federal University, 89 Kazanskaya str., Elabuga, 423600, Russia

    L. M. Kozhevnikova

Authors
  1. L. M. Kozhevnikova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. M. Kozhevnikova.

Additional information

Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 1, pp. 30–45.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kozhevnikova, L.M. Equivalence of Entropy and Renormalized Solutions of Anisotropic Elliptic Problem in Unbounded Domains with Measure Data. Russ Math. 64, 25–39 (2020). https://doi.org/10.3103/S1066369X20010041

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X20010041

Keywords

Search

Navigation