Skip to main content
SpringerLink
Log in

On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with L 1-data

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this article, we deal with a class of degenerate, nonlinear, elliptic fourth-order equations in divergence form with coefficients satisfying a strengthened ellipticity condition and right-hand sides of the class L 1 depending on the unknown function. We consider the Dirichlet problem for equations of the given class and prove the existence of solutions of this problem bounded on the sets where the behavior of the data of the problem and the weighted functions involved is sufficiently regular.

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. P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J. L. Vazquez, “An L 1-theory of existence and uniqueness of solutions of nonlinear elliptic equations,” Ann. Scu. Norm. Sup. Pisa Cl. Sci., 22, 241–273 (1995).

    MATH  Google Scholar 

  2. A. A. Kovalevskii, “Entropy solutions of the Dirichlet problem for a class of nonlinear elliptic fourth-order equations with right-hand sides in L 1,” Izv.: Math., 65, 231–283 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Kovalevsky and F. Nicolosi, “Boundedness of solutions of variational inequalities with nonlinear degenerated elliptic operators of high order,” Appl. Anal., 65, 225–249 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Kovalevsky and F. Nicolosi, “On the Hölder continuity of solutions of equations and variational inequalities with degenerate nonlinear high order operators,” in: Problemi attuali dell’analisi e della fisica matematica. Atti del 2 o Simp. Int. dedicato alla memoria del Prof. Gaetano Fichera, Aracne Editrice, Roma (2000), pp. 205–220.

    Google Scholar 

  5. A. Kovalevsky and F. Nicolosi, “Solvability of Dirichlet problem for a class of degenerate nonlinear high-order equations with L 1-data,” Nonlinear Anal., Theory Methods Appl., 47, 435–446 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Kovalevsky and F. Nicolosi, “Entropy solutions of Dirichlet problem for a class of degenerate anisotropic fourth-order equations with L 1-right-hand sides,” Nonlinear Anal., Theory Methods Appl., 50, 581–619 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Kovalevsky and F. Nicolosi, “Existence of solutions of some degenerate nonlinear elliptic fourth-order equations with L 1-data,” Appl. Anal., 81, 905–914 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  8. J. L. Lions, Quelques Méthods de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris (1969).

    Google Scholar 

  9. F. Nicolosi and I. V. Skrypnik, “Nirenberg-Gagliardo interpolation inequality and regularity of solutions of nonlinear higher order equations,” Topol. Methods Nonlinear Anal., 7, 327–347 (1996).

    MathSciNet  MATH  Google Scholar 

  10. I. V. Skrypnik, “High-order quasilinear elliptic equations with continuous generalized solutions,” Differ. Uravn., 14, 786–795 (1978).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Rosa Luxemburg St. 74, 83114, Donetsk, Ukraine

    A. A. Kovalevsky & F. Nicolosi

Authors
  1. A. A. Kovalevsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Nicolosi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Kovalevsky.

Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 99–112, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kovalevsky, A.A., Nicolosi, F. On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with L 1-data. J Math Sci 150, 2358–2368 (2008). https://doi.org/10.1007/s10958-008-0135-8

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-0135-8

Keywords

Search

Navigation