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Статья
2020

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


L. M. KozhevnikovaL. M. Kozhevnikova
Русская математика
https://doi.org/10.3103/S1066369X20010041
Abstract / Full Text

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.

Author information
  • Sterlitamak Branch of Bashkir State University, 31 Lenin Ave., Sterlitamak, 453103, RussiaL. M. Kozhevnikova
  • Elabuga Branch of Kazan Federal University, 89 Kazanskaya str., Elabuga, 423600, RussiaL. M. Kozhevnikova
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).
  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).
  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).
  4. Malusa, A., Porzio, M.M. “Renormalized solutions to elliptic equations with measure data in unbounded domains”, Nonlinear Anal. 67 (8), 2370–2389 (2007).
  5. Zhikov, V.V. “Variation problems and non-lineary elliptic equations with non-standard growth condi¬tions”, Probl. mathem. analys. 54, 23–112 (2011).
  6. Alhutov, Yu.A. “Hölder continuity of p(x)-harmonic functions”, Sb. Math. 196 (2), 147–171 (2005).
  7. Diening, L., Harjulehto, P., Hästö, P.M., Růžička, M. Lebesgue and Sobolev spaces with variable exponents (Springer, New York, 2011).
  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).
  9. Zhang, C. “Entropy solutions to nonlinear elliptic equations with variable exponents”, Electron. J. Diff. Equat. 2014 (92), 1–14 (2014).
  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).
  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).
  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).
  13. Konaté, I., Ouaro, S. “Good Radon measure for anisotropic problems with variable exponent”, Electron. J. Diff. Equat. 2016 (221), 1–19 (2016).
  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).
  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).
  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).
  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).
  18. Kozhevnikova, L.M. “Entropy and renormalized solutions of anisotropic elliptic equations with variable nonlinearity exponents”, Sb. Math. 210 (3), 417–446 (2019).
  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).
  20. Fan, X. “Anisotropic variable exponent Sobolev spaces and p(x)-Laplacian equations”, Complex Var. Elliptic Equat. 56 (7-9), 623–642 (2011).
  21. Benboubker, M.B., Azroul, E., Barbara, A. “Quasilinear elliptic problems with nonstandard growths”, Electron. J. Diff. Equat. 2011 (62), 1–16 (2011).