Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2010, Article ID 536236, 33 pages
http://dx.doi.org/10.1155/2010/536236
Research Article

Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters

1Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle, Germany
2Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France

Received 18 September 2009; Accepted 23 November 2009

Academic Editor: Thomas Bartsch

Copyright © 2010 Siegfried Carl and Dumitru Motreanu. 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. S. Carl and D. Motreanu, “Constant-sign and sign-changing solutions of a nonlinear eigenvalue problem involving the p-Laplacian,” Differential and Integral Equations, vol. 20, no. 3, pp. 309–324, 2007. View at Google Scholar · View at MathSciNet
  2. S. Carl and D. Motreanu, “Multiple solutions of nonlinear elliptic hemivariational problems,” Pacific Journal of Aplied Mathematics, vol. 1, no. 4, pp. 39–59, 2008. View at Google Scholar
  3. S. Carl and D. Motreanu, “Multiple and sign-changing solutions for the multivalued p-Laplacian equation,” Mathematische Nachrichten. In press.
  4. S. Carl and D. Motreanu, “Sign-changing and extremal constant-sign solutions of nonlinear elliptic problems with supercritical nonlinearities,” Communications on Applied Nonlinear Analysis, vol. 14, no. 4, pp. 85–100, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. S. Carl and D. Motreanu, “Constant-sign and sign-changing solutions for nonlinear eigenvalue problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 9, pp. 2668–2676, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. E. H. Papageorgiou and N. S. Papageorgiou, “A multiplicity theorem for problems with the p-Laplacian,” Journal of Functional Analysis, vol. 244, no. 1, pp. 63–77, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  7. D. Averna, S. A. Marano, and D. Motreanu, “Multiple solutions for a Dirichlet problem with p-Laplacian and set-valued nonlinearity,” Bulletin of the Australian Mathematical Society, vol. 77, no. 2, pp. 285–303, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. A. Marano, G. Molica Bisci, and D. Motreanu, “Multiple solutions for a class of elliptic hemivariational inequalities,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 85–97, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  9. R. P. Agarwal, M. E. Filippakis, D. O'Regan, and N. S. Papageorgiou, “Constant sign and nodal solutions for problems with the p-Laplacian and a nonsmooth potential using variational techniques,” Boundary Value Problems, vol. 2009, Article ID 820237, 32 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  10. A. Ambrosetti and D. Lupo, “On a class of nonlinear Dirichlet problems with multiple solutions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 8, no. 10, pp. 1145–1150, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Ambrosetti and G. Mancini, “Sharp nonuniqueness results for some nonlinear problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 3, no. 5, pp. 635–645, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. T. Bartsch and Z. Liu, “Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian,” Communications in Contemporary Mathematics, vol. 6, no. 2, pp. 245–258, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  13. N. Dancer and K. Perera, “Some remarks on the Fučík spectrum of the p-Laplacian and critical groups,” Journal of Mathematical Analysis and Applications, vol. 254, no. 1, pp. 164–177, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  14. Q. Jiu and J. Su, “Existence and multiplicity results for Dirichlet problems with p-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 587–601, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  15. E. Koizumi and K. Schmitt, “Ambrosetti-Prodi-type problems for quasilinear elliptic problems,” Differential and Integral Equations, vol. 18, no. 3, pp. 241–262, 2005. View at Google Scholar · View at MathSciNet
  16. C. Li and S. Li, “Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 14–32, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Z. Liu, “Multiplicity of solutions of semilinear elliptic boundary value problems with jumping nonlinearities at zero,” Nonlinear Analysis: Theory, Methods & Applications, vol. 48, no. 7, pp. 1051–1063, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Liu, “Multiple solutions for coercive p-Laplacian equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 229–236, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  19. A. Qian and S. Li, “Multiple sign-changing solutions of an elliptic eigenvalue problem,” Discrete and Continuous Dynamical Systems. Series A, vol. 12, no. 4, pp. 737–746, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Struwe, “A note on a result of Ambrosetti and Mancini,” Annali di Matematica Pura ed Applicata, vol. 131, pp. 107–115, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, vol. 34 of Results in Mathematics and Related Areas (3), Springer, Berlin, Germany, 2nd edition, 1996. View at MathSciNet
  22. Z. Zhang and S. Li, “On sign-changing and multiple solutions of the p-Laplacian,” Journal of Functional Analysis, vol. 197, no. 2, pp. 447–468, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Z. Zhang, J. Chen, and S. Li, “Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-Laplacian,” Journal of Differential Equations, vol. 201, no. 2, pp. 287–303, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. Z. Zhang and X. Li, “Sign-changing solutions and multiple solutions theorems for semilinear elliptic boundary value problems with a reaction term nonzero at zero,” Journal of Differential Equations, vol. 178, no. 2, pp. 298–313, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Z. Jin, “Multiple solutions for a class of semilinear elliptic equations,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3659–3667, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. D. Motreanu, V. V. Motreanu, and N. S. Papageorgiou, “Multiple nontrivial solutions for nonlinear eigenvalue problems,” Proceedings of the American Mathematical Society, vol. 135, no. 11, pp. 3649–3658, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  27. L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, vol. 8 of Series in Mathematical Analysis and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005. View at MathSciNet
  28. J. L. Vázquez, “A strong maximum principle for some quasilinear elliptic equations,” Applied Mathematics and Optimization, vol. 12, no. 3, pp. 191–202, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. S. Carl, V. K. Le, and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2007. View at MathSciNet
  30. A. Anane, “Etude des valeurs propres et de la résonance pour l'opérateur p-Laplacien,” Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, vol. 305, no. 16, pp. 725–728, 1987. View at Google Scholar
  31. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986. View at MathSciNet
  32. M. Cuesta, D. de Figueiredo, and J.-P. Gossez, “The beginning of the Fučik spectrum for the p-Laplacian,” Journal of Differential Equations, vol. 159, no. 1, pp. 212–238, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. T. Bartsch, Z. Liu, and T. Weth, “Nodal solutions of a p-Laplacian equation,” Proceedings of the London Mathematical Society, vol. 91, no. 1, pp. 129–152, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  34. D. Motreanu, V. V. Motreanu, and N. S. Papageorgiou, “A unified approach for multiple constant sign and nodal solutions,” Advances in Differential Equations, vol. 12, no. 12, pp. 1363–1392, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, vol. 29 of Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
  36. L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, vol. 9 of Series in Mathematical Analysis and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2006. View at MathSciNet
  37. F. H. Clarke, Optimization and Nonsmooth Analysis, vol. 5 of Classics in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 2nd edition, 1990. View at MathSciNet
  38. S. Carl and D. Motreanu, “General comparison principle for quasilinear elliptic inclusions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 1105–1112, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. D. Motreanu and N. S. Papageorgiou, “Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 8, pp. 1211–1234, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. D. Motreanu, V. V. Motreanu, and D. Paşca, “A version of Zhong's coercivity result for a general class of nonsmooth functionals,” Abstract and Applied Analysis, vol. 7, no. 11, pp. 601–612, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. K. C. Chang, “Variational methods for nondifferentiable functionals and their applications to partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 80, no. 1, pp. 102–129, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. J.-N. Corvellec, “Morse theory for continuous functionals,” Journal of Mathematical Analysis and Applications, vol. 196, no. 3, pp. 1050–1072, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. P. Winkert, Comparison principles and multiple solutions for nonlinear elliptic problems, Ph.D. thesis, Institute of Mathematics, University of Halle, Halle, Germany, 2009.