Skip to main content

Advertisement

You are viewing the new article page. Let us know what you think. Return to old version

Research Article | Open | Published:

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

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. 1.

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

  2. 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

  3. 3.

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

  4. 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.

  5. 5.

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

  6. 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

  7. 7.

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

  8. 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

  9. 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.

  10. 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

  11. 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

  12. 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.

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. 17.

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

Download references

Author information

Correspondence to Mihai Mihăilescu.

Rights and permissions

Reprints and Permissions

About this article

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Weak Solution
  • Functional Equation