Skip to main content
Boundary Value Problems
Download PDF
  • Research Article
  • Open access
  • Published:

Existence and multiplicity of weak solutions for a class of degenerate nonlinear elliptic equations

Boundary Value Problems volume 2006, Article number: 41295 (2006) Cite this article

Abstract

The goal of this paper is to study the existence and the multiplicity of non-trivial weak solutions for some degenerate nonlinear elliptic equations on the whole space. The solutions will be obtained in a subspace of the Sobolev space. The proofs rely essentially on the Mountain Pass theorem and on Ekeland's Variational principle.

[1234567891011121314151617]

References

  1. Alves CO, Gonçalves JV, Miyagaki OH:On elliptic equations in with critical exponents. Electronic Journal of Differential Equations 1996,1996(9):1-11.

    Google Scholar 

  2. Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. Journal of Functional Analysis 1973,14(4):349-381. 10.1016/0022-1236(73)90051-7

    Article  MathSciNet  MATH  Google Scholar 

  3. Brezis H: Analyse Fonctionnelle. Théorie et Applications, Collection of Applied Mathematics for the Master's Degree. Masson, Paris; 1983.

    Google Scholar 

  4. De Nápoli P, Mariani MC:Mountain pass solutions to equations of-Laplacian type. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 2003,54(7):1205-1219.

    Article  MathSciNet  MATH  Google Scholar 

  5. Díaz JI: Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic Equations, Research Notes in Mathematics. Volume 106. Pitman, Massachusetts; 1985.

    Google Scholar 

  6. do Ó JMB: Existence of solutions for quasilinear elliptic equations. Journal of Mathematical Analysis and Applications 1997,207(1):104-126. 10.1006/jmaa.1997.5270

    Article  MathSciNet  MATH  Google Scholar 

  7. do Ó JMB:Solutions to perturbed eigenvalue problems of the-Laplacian in. Electronic Journal of Differential Equations 1997,1997(11):1-15.

    Google Scholar 

  8. Ekeland I: On the variational principle. Journal of Mathematical Analysis and Applications 1974,47(2):324-353. 10.1016/0022-247X(74)90025-0

    Article  MathSciNet  MATH  Google Scholar 

  9. Gonçalves JV, Miyagaki OH:Multiple positive solutions for semilinear elliptic equations in involving subcritical exponents. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1998,32(1):41-51.

    Article  MathSciNet  MATH  Google Scholar 

  10. Mihăilescu M, Rădulescu V: Ground state solutions of non-linear singular Schrödinger equations with lack of compactness. Mathematical Methods in the Applied Sciences 2003,26(11):897-906. 10.1002/mma.403

    Article  MathSciNet  MATH  Google Scholar 

  11. Motreanu D, Rădulescu V: Eigenvalue problems for degenerate nonlinear elliptic equations in anisotropic media. Boundary Value Problems 2005,2005(2):107-127. 10.1155/BVP.2005.107

    Article  MATH  Google Scholar 

  12. Pflüger K:Existence and multiplicity of solutions to a-Laplacian equation with nonlinear boundary condition. Electronic Journal of Differential Equations 1998,1998(10):1-13.

    Google Scholar 

  13. Rabinowitz PH: On a class of nonlinear Schrödinger equations. Zeitschrift für Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathématiques et de Physique Appliquées 1992,43(2):270-291.

    MathSciNet  MATH  Google Scholar 

  14. Rădulescu V, Smets D: Critical singular problems on infinite cones. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 2003,54(6):1153-1164.

    Article  MathSciNet  MATH  Google Scholar 

  15. Şt. Cîrstea F, Rădulescu V: Multiple solutions of degenerate perturbed elliptic problems involving a subcritical Sobolev exponent. Topological Methods in Nonlinear Analysis 2000,15(2):283-300.

    MathSciNet  Google Scholar 

  16. Tarantello G: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1992,9(3):281-304.

    MathSciNet  MATH  Google Scholar 

  17. Willem M: Analyse harmonique réelle, Methods Collection. Hermann, Paris; 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Craiova, Craiova, 200 585, Romania

    Mihai Mihăilescu

Authors
  1. Mihai Mihăilescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mihai Mihăilescu.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Mihăilescu, M. Existence and multiplicity of weak solutions for a class of degenerate nonlinear elliptic equations. Bound Value Probl 2006, 41295 (2006). https://doi.org/10.1155/BVP/2006/41295

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/BVP/2006/41295

Keywords