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  • Research Article
  • Open Access

Superlinear Equations Involving Nonlinearities Limited by Asymptotically Homogeneous Functions

Journal of Inequalities and Applications20072007:058363

  • Received: 24 August 2006
  • Accepted: 28 March 2007
  • Published:


We obtain a solution of the quasilinear equation in , , on . Here the nonlinearity is superlinear at zero, and it is located near infinity between two functions that belong to a class of functions where the Ambrosetti-Rabinowitz condition is satisfied. More precisely, we consider the class of functions that are asymptotically homogeneous of index .


  • Homogeneous Function
  • Quasilinear Equation
  • Nonlinearity Limited
  • Asymptotically Homogeneous
  • Equation Involve


Authors’ Affiliations

Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7–D, Arica, 1000007, Chile
Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, Campina Grande, PR, 58109-900, Brazil
Departamento de Matemáticas y Ciencias de la Computación, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, 9170022, Chile


  1. Díaz JI: Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic Equations, Research Notes in Mathematics. Volume 106. Pitman, Boston, Mass, USA; 1985:vii+323.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)90051-7MathSciNetView ArticleMATHGoogle Scholar
  3. de Figueiredo DG, Gossez J-P, Ubilla P: Local superlinearity and sublinearity for indefinite semilinear elliptic problems. Journal of Functional Analysis 2003,199(2):452–467. 10.1016/S0022-1236(02)00060-5MathSciNetView ArticleMATHGoogle Scholar
  4. Gidas B, Spruck J: Global and local behavior of positive solutions of nonlinear elliptic equations. Communications on Pure and Applied Mathematics 1981,34(4):525–598. 10.1002/cpa.3160340406MathSciNetView ArticleMATHGoogle Scholar
  5. 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.2001.4029MathSciNetView ArticleMATHGoogle Scholar
  6. de Figueiredo DG, Yang J: On a semilinear elliptic problem without (PS) condition. Journal of Differential Equations 2003,187(2):412–428. 10.1016/S0022-0396(02)00055-4MathSciNetView ArticleMATHGoogle Scholar
  7. 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.2003.10.021MathSciNetView ArticleMATHGoogle Scholar
  8. García-Huidobro M, Manásevich R, Ubilla P: Existence of positive solutions for some Dirichlet problems with an asymptotically homogeneous operator. Electronic Journal of Differential Equations 1995,1995(10):1–22.Google Scholar
  9. Resnick SI: Extreme Values, Regular Variation, and Point Processes, Applied Probability. A Series of the Applied Probability Trust. Volume 4. Springer, New York, NY, USA; 1987:xii+320.Google Scholar
  10. Seneta E: Regularly Varying Functions, Lecture Notes in Mathematics. Volume 508. Springer, Berlin, Germany; 1976:v+112.View ArticleGoogle Scholar
  11. Trudinger NS: On Harnack type inequalities and their application to quasilinear elliptic equations. Communications on Pure and Applied Mathematics 1967,20(4):721–747. 10.1002/cpa.3160200406MathSciNetView ArticleMATHGoogle Scholar
  12. 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/BF02392645MathSciNetView ArticleMATHGoogle Scholar
  13. Lieberman GM: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Analysis 1988,12(11):1203–1219. 10.1016/0362-546X(88)90053-3MathSciNetView ArticleMATHGoogle Scholar


© Sebastiáan Lorca et al. 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.