Boundary Value Problems

Impact Factor 0.819

Boundary Value Problems Cover Image
Research Article
Open Access

Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations

Boundary Value Problems20072007:016407

https://doi.org/10.1155/2007/16407

©  Z. Yang and B. Xu 2007

Received: 29 June 2006

Accepted: 17 October 2006

Published: 9 January 2007

Abstract

We consider the problem where is not identically zero. Under the condition that satisfies (H), we show that there exists such that the above-mentioned equation admits at least one solution for all . This extends the results of Laplace equation to the case of -Laplace equation.

[12345678910111213141516]

Authors’ Affiliations

(1)
Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University

References

  1. Herrero MA, Vázquez JL: On the propagation properties of a nonlinear degenerate parabolic equation. Communications in Partial Differential Equations 1982,7(12):1381–1402. 10.1080/03605308208820255MATHMathSciNetView ArticleGoogle Scholar
  2. Esteban JR, Vázquez JL: On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Analysis 1986,10(11):1303–1325. 10.1016/0362-546X(86)90068-4MATHMathSciNetView ArticleGoogle Scholar
  3. Yang Z: Existence of positive bounded entire solutions for quasilinear elliptic equations. Applied Mathematics and Computation 2004,156(3):743–754. 10.1016/j.amc.2003.06.024MATHMathSciNetView ArticleGoogle Scholar
  4. Guedda M, Véron L: Local and global properties of solutions of quasilinear elliptic equations. Journal of Differential Equations 1988,76(1):159–189. 10.1016/0022-0396(88)90068-XMATHMathSciNetView ArticleGoogle Scholar
  5. Guo ZM: Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations. Applicable Analysis 1992,47(2–3):173–189.MATHMathSciNetView ArticleGoogle Scholar
  6. Guo ZM: Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems. Nonlinear Analysis 1992,18(10):957–971. 10.1016/0362-546X(92)90132-XMATHMathSciNetView ArticleGoogle Scholar
  7. Guo ZM, Webb JRL: Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1994,124(1):189–198. 10.1017/S0308210500029280MATHMathSciNetView ArticleGoogle Scholar
  8. Lu Q, Yang Z, Twizell EH: Existence of entire explosive positive solutions of quasi-linear elliptic equations. Applied Mathematics and Computation 2004,148(2):359–372. 10.1016/S0096-3003(02)00852-4MATHMathSciNetView ArticleGoogle Scholar
  9. Bognár G, Drábek P: The -Laplacian equation with superlinear and supercritical growth, multiplicity of radial solutions. Nonlinear Analysis 2005,60(4):719–728. 10.1016/j.na.2004.09.047MATHMathSciNetView ArticleGoogle Scholar
  10. Prashanth S, Sreenadh K: Multiplicity of positive solutions for -Laplace equation with superlinear-type nonlinearity. Nonlinear Analysis 2004,56(6):867–878. 10.1016/j.na.2003.10.026MATHMathSciNetView ArticleGoogle Scholar
  11. Brezis H, Kamin S: Sublinear elliptic equations in . Manuscripta Mathematica 1992,74(1):87–106. 10.1007/BF02567660MATHMathSciNetView ArticleGoogle Scholar
  12. Ambrosetti A, Brezis H, Cerami G: Combined effects of concave and convex nonlinearities in some elliptic problems. Journal of Functional Analysis 1994,122(2):519–543. 10.1006/jfan.1994.1078MATHMathSciNetView ArticleGoogle Scholar
  13. Brezis H, Oswald L: Remarks on sublinear elliptic equations. Nonlinear Analysis 1986,10(1):55–64. 10.1016/0362-546X(86)90011-8MATHMathSciNetView ArticleGoogle Scholar
  14. Bartsch T, Willem M: On an elliptic equation with concave and convex nonlinearities. Proceedings of the American Mathematical Society 1995,123(11):3555–3561. 10.1090/S0002-9939-1995-1301008-2MATHMathSciNetView ArticleGoogle Scholar
  15. Ye D, Zhou F: Invariant criteria for existence of bounded positive solutions. Discrete and Continuous Dynamical Systems. Series A 2005,12(3):413–424.MATHMathSciNetGoogle Scholar
  16. El Mabrouk K: Entire bounded solutions for a class of sublinear elliptic equations. Nonlinear Analysis 2004,58(1–2):205–218. 10.1016/j.na.2004.01.004MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Z. Yang and B. Xu 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.