Skip to content

Advertisement

Boundary Value Problems
Boundary Value Problems Cover Image
  • Research Article
  • Open Access

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

Boundary Value Problems20062006:47275

https://doi.org/10.1155/BVP/2006/47275

©  Wang and Tang 2006

  • Received: 16 May 2006
  • Accepted: 6 July 2006
  • Published:

Abstract

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 .

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Variant Version

[1234567891011121314151617181920]

Authors’ Affiliations

(1)
Department of Mathematics, School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

References

  1. 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
  2. Brézis H, Nirenberg L: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Communications on Pure and Applied Mathematics 1983,36(4):437-477. 10.1002/cpa.3160360405MathSciNetView ArticleMATHGoogle Scholar
  3. Costa DG, Magalhães CA: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Analysis 1994,23(11):1401-1412. 10.1016/0362-546X(94)90135-XMathSciNetView ArticleMATHGoogle Scholar
  4. Costa DG, Miyagaki OH:Nontrivial solutions for perturbations of the -Laplacian on unbounded domains. Journal of Mathematical Analysis and Applications 1995,193(3):737-755. 10.1006/jmaa.1995.1264MathSciNetView ArticleMATHGoogle Scholar
  5. Goncalves JV, Meira S: On a class of semilinear elliptic problems near critical growth. International Journal of Mathematics and Mathematical Sciences 1998,21(2):321-330. 10.1155/S0161171298000441MathSciNetView ArticleMATHGoogle Scholar
  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 1999,129(4):787-809. 10.1017/S0308210500013147MathSciNetView ArticleMATHGoogle Scholar
  7. Ladyzhenskaya OA, Ural'tseva NN: Linear and Quasilinear Elliptic Equations. Academic Press, New York; 1968:xviii+495.MATHGoogle Scholar
  8. Li G, Zhou H-S:Asymptotically linear Dirichlet problem for the -Laplacian. Nonlinear Analysis, Series A: Theory Methods 2001,43(8):1043-1055. 10.1016/S0362-546X(99)00243-6View ArticleMATHMathSciNetGoogle Scholar
  9. Liu SB, Li SJ: Infinitely many solutions for a superlinear elliptic equation. Acta Mathematica Sinica 2003,46(4):625-630.MathSciNetMATHGoogle Scholar
  10. Perera K, Schechter M: Semilinear elliptic equations having asymptotic limits at zero and infinity. Abstract and Applied Analysis 1999,4(4):231-242. 10.1155/S1085337599000159MathSciNetView ArticleMATHGoogle Scholar
  11. Qian A: Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem. Boundary Value Problems 2005, (3):329-335.Google Scholar
  12. Rabinowitz PH: Minimax methods and their application to partial differential equations. In Seminar on Nonlinear Partial Differential Equations (Berkeley, Calif, 1983), Math. Sci. Res. Inst. Publ.. Volume 2. Springer, New York; 1984:307-320.View ArticleGoogle Scholar
  13. Schechter M: A variation of the mountain pass lemma and applications. Journal of the London Mathematical Society. Second Series 1991,44(3):491-502. 10.1112/jlms/s2-44.3.491MathSciNetView ArticleMATHGoogle Scholar
  14. Schechter M, Zou W: Superlinear problems. Pacific Journal of Mathematics 2004,214(1):145-160. 10.2140/pjm.2004.214.145MathSciNetView ArticleMATHGoogle Scholar
  15. Szulkin A, Zou W: Homoclinic orbits for asymptotically linear Hamiltonian systems. Journal of Functional Analysis 2001,187(1):25-41. 10.1006/jfan.2001.3798MathSciNetView ArticleMATHGoogle Scholar
  16. Tolksdorf P: Regularity for a more general class of quasilinear elliptic equations. Journal of Differential Equations 1984,51(1):126-150. 10.1016/0022-0396(84)90105-0MathSciNetView ArticleMATHGoogle Scholar
  17. Vázquez JL: A strong maximum principle for some quasilinear elliptic equations. Applied Mathematics and Optimization 1984,12(3):191-202.MathSciNetView ArticleMATHGoogle Scholar
  18. Zhou H-S: Existence of asymptotically linear Dirichlet problem. Nonlinear Analysis. Series A: Theory and Methods 2001,44(7):909-918. 10.1016/S0362-546X(99)00314-4View ArticleMATHMathSciNetGoogle Scholar
  19. Zhou H-S: An application of a mountain pass theorem. Acta Mathematica Sinica 2002,18(1):27-36. 10.1007/s101140100147MathSciNetView ArticleMATHGoogle Scholar
  20. Zou W: Variant fountain theorems and their applications. Manuscripta Mathematica 2001,104(3):343-358. 10.1007/s002290170032MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Wang and Tang 2006

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.

Advertisement