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  • 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

  • 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

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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

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