Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 749059, 14 pages
http://dx.doi.org/10.1155/2012/749059
Research Article

Multiplicity of Solutions for a Class of Fourth-Order Elliptic Problems with Asymptotically Linear Term

School of Mathematics and Statistics, Hubei Engineering University, Hubei, Xiaogan 432000, China

Received 10 January 2012; Accepted 5 April 2012

Academic Editor: Turgut Öziş

Copyright © 2012 Qiong Liu and Dengfeng Lü. 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.

Linked References

  1. A. C. Lazer and P. J. McKenna, “Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis,” SIAM Review, vol. 32, no. 4, pp. 537–578, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. A. C. Lazer and P. J. McKenna, “Global bifurcation and a theorem of Tarantello,” Journal of Mathematical Analysis and Applications, vol. 181, no. 3, pp. 648–655, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. P. J. McKenna and W. Reichel, “Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry,” Electronic Journal of Differential Equations, vol. 37, pp. 1–13, 2003. View at Google Scholar · View at Zentralblatt MATH
  4. G. Tarantello, “A note on a semilinear elliptic problem,” Differential and Integral Equations, vol. 5, no. 3, pp. 561–565, 1992. View at Google Scholar · View at Zentralblatt MATH
  5. A. M. Micheletti and A. Pistoia, “Multiplicity results for a fourth-order semilinear elliptic problem,” Nonlinear Analysis, vol. 31, no. 7, pp. 895–908, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. A. M. Micheletti and A. Pistoia, “Nontrivial solutions for some fourth order semilinear elliptic problems,” Nonlinear Analysis, vol. 34, no. 4, pp. 509–523, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. G. Xu and J. Zhang, “Existence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 633–640, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. X. Liu and Y. Huang, “On sign-changing solution for a fourth-order asymptotically linear elliptic problem,” Nonlinear Analysis, vol. 72, no. 5, pp. 2271–2276, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. Y. An and R. Liu, “Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation,” Nonlinear Analysis, vol. 68, no. 11, pp. 3325–3331, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Y. Yang and J. Zhang, “Existence of solutions for some fourth-order nonlinear elliptic problems,” Journal of Mathematical Analysis and Applications, vol. 351, no. 1, pp. 128–137, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. J. Zhang and S. Li, “Multiple nontrivial solutions for some fourth-order semilinear elliptic problems,” Nonlinear Analysis, vol. 60, no. 2, pp. 221–230, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Z. Jihui, “Existence results for some fourth-order nonlinear elliptic problems,” Nonlinear Analysis, vol. 45, pp. 29–36, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J. Zhou and X. Wu, “Sign-changing solutions for some fourth-order nonlinear elliptic problems,” Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 542–558, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. Zhang and Z. Wei, “Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin,” Journal of Mathematical Analysis and Applications, vol. 383, no. 2, pp. 291–306, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Y. Liu and Z. Wang, “Biharmonic equations with asymptotically linear nonlinearities,” Acta Mathematica Scientia. Series B, vol. 27, no. 3, pp. 549–560, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. D. Lü, “Multiple solutions for a class of biharmonic elliptic systems with Sobolev critical exponent,” Nonlinear Analysis, vol. 74, no. 17, pp. 6371–6382, 2011. View at Publisher · View at Google Scholar
  17. Y. Yin and X. Wu, “High energy solutions and nontrivial solutions for fourth-order elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 375, no. 2, pp. 699–705, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. P. Mironescu and V. D. Rădulescu, “The study of a bifurcation problem associated to an asymptotically linear function,” Nonlinear Analysis, vol. 26, no. 4, pp. 857–875, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. P. Pucci and V. Rădulescu, “The impact of the mountain pass theory in nonlinear analysis: a mathematical survey,” Bollettino della Unione Matematica Italiana. Serie 9, vol. 3, no. 3, pp. 543–582, 2010. View at Google Scholar · View at Zentralblatt MATH
  20. V. D. Rădulescu, Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods, vol. 6 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2008.
  21. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, Conference Board of the Mathematical Sciences, Washington, DC, USA, 1986.
  22. D. G. Costa and C. A. Magalhães, “Existence results for perturbations of the p-Laplacian,” Nonlinear Analysis, vol. 24, no. 3, pp. 409–418, 1995. View at Publisher · View at Google Scholar
  23. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
  24. P. Mironescu and V. D. Rădulescu, “A multiplicity theorem for locally Lipschitz periodic functionals,” Journal of Mathematical Analysis and Applications, vol. 195, no. 3, pp. 621–637, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” vol. 14, pp. 349–381, 1973. View at Google Scholar · View at Zentralblatt MATH
  26. G. Li and H.-S. Zhou, “Multiple solutions to p-Laplacian problems with asymptotic nonlinearity as up-1 at infinity,” Journal of the London Mathematical Society. Second Series, vol. 65, no. 1, pp. 123–138, 2002. View at Publisher · View at Google Scholar
  27. G. Li and H.-S. Zhou, “Asymptotically linear Dirichlet problem for the p-Laplacian,” Nonlinear Analysis, vol. 43, pp. 1043–1055, 2001. View at Publisher · View at Google Scholar
  28. C. A. Stuart and H. S. Zhou, “Applying the mountain pass theorem to an asymptotically linear elliptic equation on RN,” Communications in Partial Differential Equations, vol. 24, no. 9-10, pp. 1731–1758, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH