Skip to main content

Existence and Multiplicity Results for Degenerate Elliptic Equations with Dependence on the Gradient

Abstract

We study the existence of positive solutions for a class of degenerate nonlinear elliptic equations with gradient dependence. For this purpose, we combine a blowup argument, the strong maximum principle, and Liouville-type theorems to obtain a priori estimates.

[12345678910111213141516171819]

References

  1. 1.

    Dong W: A priori estimates and existence of positive solutions for a quasilinear elliptic equation. Journal of the London Mathematical Society 2005,72(3):645–662. 10.1112/S0024610705006848

    MATH  MathSciNet  Article  Google Scholar 

  2. 2.

    Ruiz D: A priori estimates and existence of positive solutions for strongly nonlinear problems. Journal of Differential Equations 2004,199(1):96–114. 10.1016/j.jde.2003.10.021

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Azizieh C, Clément P: A priori estimates and continuation methods for positive solutions of -Laplace equations. Journal of Differential Equations 2002,179(1):213–245. 10.1006/jdeq.2001.4029

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Takeuchi S: Positive solutions of a degenerate elliptic equation with logistic reaction. Proceedings of the American Mathematical Society 2001,129(2):433–441. 10.1090/S0002-9939-00-05723-3

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Dong W, Chen JT: Existence and multiplicity results for a degenerate elliptic equation. Acta Mathematica Sinica 2006,22(3):665–670. 10.1007/s10114-005-0696-0

    MATH  MathSciNet  Article  Google Scholar 

  6. 6.

    Rabinowitz PH: Pairs of positive solutions of nonlinear elliptic partial differential equations. Indiana University Mathematics Journal 1973/1974, 23: 173–186. 10.1512/iumj.1973.23.23014

    MathSciNet  Article  Google Scholar 

  7. 7.

    Díaz JI, Saá JE: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. [Existence and uniqueness of positive solutions of some quasilinear elliptic equations]. Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique 1987,305(12):521–524.

    MATH  Google Scholar 

  8. 8.

    García Melián J, de Lis JS: Uniqueness to quasilinear problems for the -Laplacian in radially symmetric domains. Nonlinear Analysis. Theory, Methods & Applications 2001,43(7):803–835. 10.1016/S0362-546X(99)00236-9

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    Guo Z, Zhang H: On the global structure of the set of positive solutions for some quasilinear elliptic boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 2001,46(7):1021–1037. 10.1016/S0362-546X(00)00160-7

    MATH  MathSciNet  Article  Google Scholar 

  10. 10.

    Takeuchi S, Yamada Y: Asymptotic properties of a reaction-diffusion equation with degenerate -Laplacian. Nonlinear Analysis. Theory, Methods & Applications 2000,42(1):41–61. 10.1016/S0362-546X(98)00329-0

    MATH  MathSciNet  Article  Google Scholar 

  11. 11.

    Takeuchi S: Multiplicity result for a degenerate elliptic equation with logistic reaction. Journal of Differential Equations 2001,173(1):138–144. 10.1006/jdeq.2000.3914

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Takeuchi S: Stationary profiles of degenerate problems with inhomogeneous saturation values. Nonlinear Analysis. Theory, Methods & Applications 2005,63(5–7):e1009-e1016.

    MATH  Article  Google Scholar 

  13. 13.

    Kamin S, Véron L: Flat core properties associated to the -Laplace operator. Proceedings of the American Mathematical Society 1993,118(4):1079–1085.

    MATH  MathSciNet  Google Scholar 

  14. 14.

    Serrin J, Zou H: Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Mathematica 2002,189(1):79–142. 10.1007/BF02392645

    MATH  MathSciNet  Article  Google Scholar 

  15. 15.

    Trudinger NS: On Harnack type inequalities and their application to quasilinear elliptic equations. Communications on Pure and Applied Mathematics 1967, 20: 721–747. 10.1002/cpa.3160200406

    MATH  MathSciNet  Article  Google Scholar 

  16. 16.

    Damascelli L: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1998,15(4):493–516.

    MATH  MathSciNet  Article  Google Scholar 

  17. 17.

    Vázquez JL: A strong maximum principle for some quasilinear elliptic equations. Applied Mathematics and Optimization 1984,12(3):191–202.

    MATH  MathSciNet  Article  Google Scholar 

  18. 18.

    Lieberman GM: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Analysis. Theory, Methods & Applications 1988,12(11):1203–1219. 10.1016/0362-546X(88)90053-3

    MATH  MathSciNet  Article  Google Scholar 

  19. 19.

    Amann H, López-Gómez J: A priori bounds and multiple solutions for superlinear indefinite elliptic problems. Journal of Differential Equations 1998,146(2):336–374. 10.1006/jdeq.1998.3440

    MATH  MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Leonelo Iturriaga.

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

Iturriaga, L., Lorca, S. Existence and Multiplicity Results for Degenerate Elliptic Equations with Dependence on the Gradient. Bound Value Probl 2007, 047218 (2007). https://doi.org/10.1155/2007/47218

Download citation

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Maximum Principle