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  • Research Article
  • Open Access

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

Boundary Value Problems20072007:047218

  • Received: 17 October 2006
  • Accepted: 9 February 2007
  • Published:


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.


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


Authors’ Affiliations

Departamento de Ingeniería Matemática y Centro de Modelamiento Matematico, Universidad de Chile, Casilla 170 Correo 3, Santiago, 8370459, Chile
Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7 D, Arica, 1000007, Chile


  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/S0024610705006848MATHMathSciNetView ArticleGoogle Scholar
  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.2003.10.021MATHMathSciNetView ArticleGoogle Scholar
  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.2001.4029MATHMathSciNetView ArticleGoogle Scholar
  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/S0002-9939-00-05723-3MATHMathSciNetView ArticleGoogle Scholar
  5. Dong W, Chen JT: Existence and multiplicity results for a degenerate elliptic equation. Acta Mathematica Sinica 2006,22(3):665–670. 10.1007/s10114-005-0696-0MATHMathSciNetView ArticleGoogle Scholar
  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.1973.23.23014MathSciNetView ArticleGoogle Scholar
  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.MATHGoogle Scholar
  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)00236-9MATHMathSciNetView ArticleGoogle Scholar
  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)00160-7MATHMathSciNetView ArticleGoogle Scholar
  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)00329-0MATHMathSciNetView ArticleGoogle Scholar
  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.2000.3914MATHMathSciNetView ArticleGoogle Scholar
  12. Takeuchi S: Stationary profiles of degenerate problems with inhomogeneous saturation values. Nonlinear Analysis. Theory, Methods & Applications 2005,63(5–7):e1009-e1016.MATHView ArticleGoogle Scholar
  13. Kamin S, Véron L: Flat core properties associated to the -Laplace operator. Proceedings of the American Mathematical Society 1993,118(4):1079–1085.MATHMathSciNetGoogle Scholar
  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/BF02392645MATHMathSciNetView ArticleGoogle Scholar
  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.3160200406MATHMathSciNetView ArticleGoogle Scholar
  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.MATHMathSciNetView ArticleGoogle Scholar
  17. Vázquez JL: A strong maximum principle for some quasilinear elliptic equations. Applied Mathematics and Optimization 1984,12(3):191–202.MATHMathSciNetView ArticleGoogle Scholar
  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)90053-3MATHMathSciNetView ArticleGoogle Scholar
  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.1998.3440MATHMathSciNetView ArticleGoogle Scholar


© L. Iturriaga and S. Lorca 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.