Skip to main content

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

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.

[1234567891011121314151617181920]

References

  1. 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)90051-7

    MathSciNet  Article  MATH  Google Scholar 

  2. 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.3160360405

    MathSciNet  Article  MATH  Google Scholar 

  3. 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-X

    MathSciNet  Article  MATH  Google Scholar 

  4. 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.1995.1264

    MathSciNet  Article  MATH  Google Scholar 

  5. 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/S0161171298000441

    MathSciNet  Article  MATH  Google Scholar 

  6. 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/S0308210500013147

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

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

    Google Scholar 

  8. 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)00243-6

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

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

    MathSciNet  MATH  Google Scholar 

  10. 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/S1085337599000159

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

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

  12. 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.

    Google Scholar 

  13. 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/s2-44.3.491

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Schechter M, Zou W: Superlinear problems. Pacific Journal of Mathematics 2004,214(1):145-160. 10.2140/pjm.2004.214.145

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Szulkin A, Zou W: Homoclinic orbits for asymptotically linear Hamiltonian systems. Journal of Functional Analysis 2001,187(1):25-41. 10.1006/jfan.2001.3798

    MathSciNet  Article  MATH  Google Scholar 

  16. 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)90105-0

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

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

    MathSciNet  Article  MATH  Google Scholar 

  18. 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)00314-4

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Zhou H-S: An application of a mountain pass theorem. Acta Mathematica Sinica 2002,18(1):27-36. 10.1007/s101140100147

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Zou W: Variant fountain theorems and their applications. Manuscripta Mathematica 2001,104(3):343-358. 10.1007/s002290170032

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Juan Wang.

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

Wang, J., Tang, CL. Existence and multiplicity of solutions for a class of superlinear-Laplacian equations. Bound Value Probl 2006, 47275 (2006). https://doi.org/10.1155/BVP/2006/47275

Download citation

Keywords

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