Skip to main content

Superlinear Equations Involving Nonlinearities Limited by Asymptotically Homogeneous Functions

Abstract

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.

[12345678910111213]

References

  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)90051-7

    Article  MathSciNet  MATH  Google 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)00060-5

    Article  MathSciNet  MATH  Google 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.3160340406

    Article  MathSciNet  MATH  Google 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.2001.4029

    Article  MathSciNet  MATH  Google 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)00055-4

    Article  MathSciNet  MATH  Google 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.2003.10.021

    Article  MathSciNet  MATH  Google 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.

    Book  Google 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.3160200406

    Article  MathSciNet  MATH  Google 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/BF02392645

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sebastián Lorca.

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

Lorca, S., Souto, M.A. & Ubilla, P. Superlinear Equations Involving Nonlinearities Limited by Asymptotically Homogeneous Functions. J Inequal Appl 2007, 058363 (2007). https://doi.org/10.1155/2007/58363

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/58363

Keywords