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Open Access Research Article

Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations

Zuodong Yang* and Bing Xu

Author Affiliations

Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Jiangsu Nanjing 210097, China

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Boundary Value Problems 2007, 2007:016407  doi:10.1155/2007/16407

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/016407

Received:29 June 2006
Accepted:17 October 2006
Published:9 January 2007

© 2007 Yang and Xu

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 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.


  1. Herrero, MA, Vázquez, JL: On the propagation properties of a nonlinear degenerate parabolic equation. Communications in Partial Differential Equations. 7(12), 1381–1402 (1982). Publisher Full Text OpenURL

  2. Esteban, JR, Vázquez, JL: On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Analysis. 10(11), 1303–1325 (1986). Publisher Full Text OpenURL

  3. Yang, Z: Existence of positive bounded entire solutions for quasilinear elliptic equations. Applied Mathematics and Computation. 156(3), 743–754 (2004). Publisher Full Text OpenURL

  4. Guedda, M, Véron, L: Local and global properties of solutions of quasilinear elliptic equations. Journal of Differential Equations. 76(1), 159–189 (1988). Publisher Full Text OpenURL

  5. Guo, ZM: Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations. Applicable Analysis. 47(2-3), 173–189 (1992)

  6. Guo, ZM: Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems. Nonlinear Analysis. 18(10), 957–971 (1992). Publisher Full Text OpenURL

  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. 124(1), 189–198 (1994). Publisher Full Text OpenURL

  8. Lu, Q, Yang, Z, Twizell, EH: Existence of entire explosive positive solutions of quasi-linear elliptic equations. Applied Mathematics and Computation. 148(2), 359–372 (2004). Publisher Full Text OpenURL

  9. Bognár, G, Drábek, P: The -Laplacian equation with superlinear and supercritical growth, multiplicity of radial solutions. Nonlinear Analysis. 60(4), 719–728 (2005). Publisher Full Text OpenURL

  10. Prashanth, S, Sreenadh, K: Multiplicity of positive solutions for -Laplace equation with superlinear-type nonlinearity. Nonlinear Analysis. 56(6), 867–878 (2004). Publisher Full Text OpenURL

  11. Brezis, H, Kamin, S: Sublinear elliptic equations in . Manuscripta Mathematica. 74(1), 87–106 (1992). Publisher Full Text OpenURL

  12. Ambrosetti, A, Brezis, H, Cerami, G: Combined effects of concave and convex nonlinearities in some elliptic problems. Journal of Functional Analysis. 122(2), 519–543 (1994). Publisher Full Text OpenURL

  13. Brezis, H, Oswald, L: Remarks on sublinear elliptic equations. Nonlinear Analysis. 10(1), 55–64 (1986). Publisher Full Text OpenURL

  14. Bartsch, T, Willem, M: On an elliptic equation with concave and convex nonlinearities. Proceedings of the American Mathematical Society. 123(11), 3555–3561 (1995). Publisher Full Text OpenURL

  15. Ye, D, Zhou, F: Invariant criteria for existence of bounded positive solutions. Discrete and Continuous Dynamical Systems. Series A. 12(3), 413–424 (2005)

  16. El Mabrouk, K: Entire bounded solutions for a class of sublinear elliptic equations. Nonlinear Analysis. 58(1-2), 205–218 (2004). Publisher Full Text OpenURL