SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Open Badges Research Article

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

Leonelo Iturriaga1* and Sebastian Lorca2

Author Affiliations

1 Departamento de Ingeniería Matemática y Centro de Modelamiento Matematico, Universidad de Chile, Casilla 170 Correo 3, Santiago 8370459, Chile

2 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7 D, Arica 1000007, Chile

For all author emails, please log on.

Boundary Value Problems 2007, 2007:047218  doi:10.1155/2007/47218

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/047218

Received:17 October 2006
Revisions received:2 January 2007
Accepted:9 February 2007
Published:5 April 2007

© 2007 Iturriaga and Lorca

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.

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.


  1. Dong, W: A priori estimates and existence of positive solutions for a quasilinear elliptic equation. Journal of the London Mathematical Society. 72(3), 645–662 (2005). Publisher Full Text OpenURL

  2. Ruiz, D: A priori estimates and existence of positive solutions for strongly nonlinear problems. Journal of Differential Equations. 199(1), 96–114 (2004). Publisher Full Text OpenURL

  3. Azizieh, C, Clément, P: A priori estimates and continuation methods for positive solutions of -Laplace equations. Journal of Differential Equations. 179(1), 213–245 (2002). Publisher Full Text OpenURL

  4. Takeuchi, S: Positive solutions of a degenerate elliptic equation with logistic reaction. Proceedings of the American Mathematical Society. 129(2), 433–441 (2001). Publisher Full Text OpenURL

  5. Dong, W, Chen, JT: Existence and multiplicity results for a degenerate elliptic equation. Acta Mathematica Sinica. 22(3), 665–670 (2006). Publisher Full Text OpenURL

  6. Rabinowitz, PH: Pairs of positive solutions of nonlinear elliptic partial differential equations. Indiana University Mathematics Journal. 23, 173–186 (1973/1974). Publisher Full Text OpenURL

  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. 305(12), 521–524 (1987). PubMed Abstract | Publisher Full Text OpenURL

  8. García Melián, J, de Lis, JS: Uniqueness to quasilinear problems for the -Laplacian in radially symmetric domains. Nonlinear Analysis. Theory, Methods & Applications. 43(7), 803–835 (2001). PubMed Abstract | Publisher Full Text OpenURL

  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. 46(7), 1021–1037 (2001). PubMed Abstract | Publisher Full Text OpenURL

  10. Takeuchi, S, Yamada, Y: Asymptotic properties of a reaction-diffusion equation with degenerate -Laplacian. Nonlinear Analysis. Theory, Methods & Applications. 42(1), 41–61 (2000). PubMed Abstract | Publisher Full Text OpenURL

  11. Takeuchi, S: Multiplicity result for a degenerate elliptic equation with logistic reaction. Journal of Differential Equations. 173(1), 138–144 (2001). Publisher Full Text OpenURL

  12. Takeuchi, S: Stationary profiles of degenerate problems with inhomogeneous saturation values. Nonlinear Analysis. Theory, Methods & Applications. 63(5–7), e1009–e1016 (2005). PubMed Abstract | Publisher Full Text OpenURL

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

  14. Serrin, J, Zou, H: Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Mathematica. 189(1), 79–142 (2002). Publisher Full Text OpenURL

  15. Trudinger, NS: On Harnack type inequalities and their application to quasilinear elliptic equations. Communications on Pure and Applied Mathematics. 20, 721–747 (1967). Publisher Full Text OpenURL

  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. 15(4), 493–516 (1998)

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

  18. Lieberman, GM: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Analysis. Theory, Methods & Applications. 12(11), 1203–1219 (1988). PubMed Abstract | Publisher Full Text OpenURL

  19. Amann, H, López-Gómez, J: A priori bounds and multiple solutions for superlinear indefinite elliptic problems. Journal of Differential Equations. 146(2), 336–374 (1998). Publisher Full Text OpenURL