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Three Solutions for Systems of n Fourth-Order Partial Differential Equations

  • Shapour Heidarkhani
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 47)

Abstract

In this paper, we shall establish the existence of at least three weak solutions for a class of systems of n fourth-order partial differential equations coupled with Navier boundary conditions. The technical approach is fully based on a very recent three critical points theorem.

Keywords

Three solutions Critical point Variational methods \((p_{1},\ldots,p_{n})\)-biharmonic Navier boundary value problem 

Notes

Acknowledgement

This research was in part supported by a grant from IPM (No. 90470020).

References

  1. 1.
    Afrouzi, G.A., Heidarkhani, S., O’Regan, D.: Existence of three solutions for a doubly eigenvalue fourth-order boundary value problem Taiwanese. J. Math. 15(1), 201–210 (2011)MathSciNetMATHGoogle Scholar
  2. 2.
    Bonanno, G., Di Bella, B.: A boundary value problem for fourth-order elastic beam equations. J. Math. Anal. Appl. 343, 1166–1176 (2010)CrossRefGoogle Scholar
  3. 3.
    Bonanno, G., Di Bella, B.: A fourth-order boundary value problem for a Sturm-Liouville type equation. Appl. Math. Comput. 217, 3635–3640 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bonanno, G., Di Bella, B.: Infinitely many solutions for a fourth-order elastic beam equation. Nonlinear Diff. Equat. Appl. NoDEA 18(3), 357–368 (2011). doi:10.1007/s00030-011-0099-0CrossRefMATHGoogle Scholar
  5. 5.
    Bonanno, G., Marano, S.A.: On the structure of the critical set of non-differentiable functions with a weak compactness condition. Appl. Anal. 89, 1–10 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bonanno, G., Di Bella, B., O’Regan, D.: Non-trivial solutions for nonlinear fourth-order elastic beam equations. Comput. Math. Appl. 62(4), 1862–1869 (2011). doi:10.1016/j.camwa.2011.06.029MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Candito, P., Livrea, R.: Infinitely many solutions for a nonlinear Navier boundary value problem involving the p-biharmonic. Studia Univ. Babeş-Bolyai Math. LV(4), 41–51 (2010)Google Scholar
  8. 8.
    Candito, P., Molica Bisci, G.: Multiple solutions for a Navier boundary value problem involving the p-biharmonic. Discrete Contin. Dyn. Syst. Ser. S 5(4) (2012). doi:10.3934/dsdss.2012.5.741Google Scholar
  9. 9.
    Graef, J.R., Heidarkhani, S., Kong, L.: Multiple solutions for a class of \((p_{1},\ldots,p_{n})\)-biharmonic systems (preprint)Google Scholar
  10. 10.
    Heidarkhani, S., Tian, Y., Tang, C.-L.: Existence of three solutions for a class of \((p_{1},\ldots,p_{n})\)-biharmonic systems with Navier boundary conditions. Ann. Polon. Math. (to appear)Google Scholar
  11. 11.
    Lazer, A.C., McKenna, P.J.: Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis. SIAM Rev. 32, 537–578 (1990)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Li, L., Tang, C.-L.: Existence of three solutions for (p,q)-biharmonic systems. Nonlinear Anal. 73, 796–805 (2010)Google Scholar
  13. 13.
    Li, C., Tang, C.-L.: Three solutions for a Navier boundary value problem involving the p-biharmonic. Nonlinear Anal. 72,1339–1347 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Liu, H., Su, N.: Existence of three solutions for a p-biharmonic problem. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 15(3), 445–452 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ricceri, B.: Existence of three solutions for a class of elliptic eigenvalue problem. Math. Comput. Model. 32, 1485–1494 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ricceri, B.: On a three critical points theorem. Arch. Math. (Basel) 75, 220–226 (2000)Google Scholar
  17. 17.
    Ricceri, B.: A three critical points theorem revisited. Nonlinear Anal. 70, 3084–3089 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Zeidler, E.: Nonlinear functional analysis and its applications, vol. II. Springer, Berlin (1985)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsRazi UniversityKermanshahIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

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