Open Access Research Article

Existence and multiplicity of solutions for a class of superlinear -Laplacian equations

Juan Wang* and Chun-Lei Tang

Author Affiliations

Department of Mathematics, School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

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Boundary Value Problems 2006, 2006:47275 doi:10.1155/BVP/2006/47275


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2006/1/47275


Received:16 May 2006
Revisions received:5 July 2006
Accepted:6 July 2006
Published:21 November 2006

© 2006 Wang and Tang

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.

By a variant version of mountain pass theorem, the existence and multiplicity of solutions are obtained for a class of superlinear -Laplacian equations: . In this paper, we suppose neither satisfies the superquadratic condition in Ambrosetti-Rabinowitz sense nor is nondecreasing with respect to .

References

  1. Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. Journal of Functional Analysis. 14(4), 349–381 (1973). Publisher Full Text OpenURL

  2. Brézis, H, Nirenberg, L: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Communications on Pure and Applied Mathematics. 36(4), 437–477 (1983). Publisher Full Text OpenURL

  3. Costa, DG, Magalhães, CA: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Analysis. 23(11), 1401–1412 (1994). Publisher Full Text OpenURL

  4. Costa, DG, Miyagaki, OH: Nontrivial solutions for perturbations of the -Laplacian on unbounded domains. Journal of Mathematical Analysis and Applications. 193(3), 737–755 (1995). Publisher Full Text OpenURL

  5. Goncalves, JV, Meira, S: On a class of semilinear elliptic problems near critical growth. International Journal of Mathematics and Mathematical Sciences. 21(2), 321–330 (1998). Publisher Full Text OpenURL

  6. Jeanjean, L: On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on . Proceedings of the Royal Society of Edinburgh. Section A. Mathematics. 129(4), 787–809 (1999). Publisher Full Text OpenURL

  7. Ladyzhenskaya, OA, Ural'tseva, NN: Linear and Quasilinear Elliptic Equations,p. xviii+495. Academic Press, New York (1968)

  8. Li, G, Zhou, H-S: Asymptotically linear Dirichlet problem for the -Laplacian. Nonlinear Analysis, Series A: Theory Methods. 43(8), 1043–1055 (2001). Publisher Full Text OpenURL

  9. Liu, SB, Li, SJ: Infinitely many solutions for a superlinear elliptic equation. Acta Mathematica Sinica. 46(4), 625–630 (2003)

  10. Perera, K, Schechter, M: Semilinear elliptic equations having asymptotic limits at zero and infinity. Abstract and Applied Analysis. 4(4), 231–242 (1999). Publisher Full Text OpenURL

  11. Qian, A: Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem. Boundary Value Problems.(3), 329–335 (2005)

  12. Rabinowitz, PH: Minimax methods and their application to partial differential equations. Seminar on Nonlinear Partial Differential Equations (Berkeley, Calif, 1983), Math. Sci. Res. Inst. Publ., pp. 307–320. Springer, New York (1984)

  13. Schechter, M: A variation of the mountain pass lemma and applications. Journal of the London Mathematical Society. Second Series. 44(3), 491–502 (1991). Publisher Full Text OpenURL

  14. Schechter, M, Zou, W: Superlinear problems. Pacific Journal of Mathematics. 214(1), 145–160 (2004). Publisher Full Text OpenURL

  15. Szulkin, A, Zou, W: Homoclinic orbits for asymptotically linear Hamiltonian systems. Journal of Functional Analysis. 187(1), 25–41 (2001). Publisher Full Text OpenURL

  16. Tolksdorf, P: Regularity for a more general class of quasilinear elliptic equations. Journal of Differential Equations. 51(1), 126–150 (1984). Publisher Full Text OpenURL

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

  18. Zhou, H-S: Existence of asymptotically linear Dirichlet problem. Nonlinear Analysis. Series A: Theory and Methods. 44(7), 909–918 (2001). Publisher Full Text OpenURL

  19. Zhou, H-S: An application of a mountain pass theorem. Acta Mathematica Sinica. 18(1), 27–36 (2002). Publisher Full Text OpenURL

  20. Zou, W: Variant fountain theorems and their applications. Manuscripta Mathematica. 104(3), 343–358 (2001). Publisher Full Text OpenURL