DOI QR코드

DOI QR Code

EXISTENCE OF THREE SOLUTIONS FOR A CLASS OF NAVIER QUASILINEAR ELLIPTIC SYSTEMS INVOLVING THE (p1, …, pn)-BIHARMONIC

Li, Lin

  • Received : 2011.03.12
  • Published : 2013.01.31

Abstract

In this paper, we establish the existence of at least three solutions to a Navier boundary problem involving the ($p_1$, ${\cdots}$, $p_n$)-biharmonic systems. We use a variational approach based on a three critical points theorem due to Ricceri [B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. 70 (2009), 3084-3089].

Keywords

($p_1$, ${\cdots}$, $p_n$)-biharmonic;Navier condition;multiple solutions;three critical points theorem

References

  1. G. A. Afrouzi and S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the ($p_1$, . . . , $p_n$)-Laplacian, Nonlinear Anal. 70 (2009), no. 1, 135-143. https://doi.org/10.1016/j.na.2007.11.038
  2. G. A. Afrouzi and S. Heidarkhani, Multiplicity results for a two-point boundary value double eigenvalue problem, Ric. Mat. 59 (2010), no. 1, 39-47. https://doi.org/10.1007/s11587-010-0072-y
  3. G. A. Afrouzi and S. Heidarkhani, Multiplicity theorems for a class of Dirichlet quasilinear elliptic systems in-volving the ($p_1$, . . . , $p_n$)-Laplacian, Nonlinear Anal. 73 (2010), no. 8, 2594-2602. https://doi.org/10.1016/j.na.2010.06.038
  4. G. A. Afrouzi, S. Heidarkhani, and D. O'Regan, Three solutions to a class of Neumann doubly eigenvalue elliptic systems driven by a ($p_1$, . . . , $p_n$)-Laplacian, Bull. Korean Math. Soc. 47 (2010), no. 6, 1235-1250. https://doi.org/10.4134/BKMS.2010.47.6.1235
  5. G. Bonanno and B. Di Bella. A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl. 343 (2008), no. 2, 1166-1176. https://doi.org/10.1016/j.jmaa.2008.01.049
  6. F. Cammaroto, A. Chinni, and B. Di Bella, Multiple solutions for a Neumann problem involving the $p({\mathit{x}})$-Laplacian, Nonlinear Anal. 71 (2009), no. 10, 4486-4492. https://doi.org/10.1016/j.na.2009.03.009
  7. J. Chabrowski and J. Marcos do O, On some fourth-order semilinear elliptic problems in ${\mathbb{R}^N}$, Nonlinear Anal. 49 (2002), no. 6, 861-884. https://doi.org/10.1016/S0362-546X(01)00144-4
  8. S. El Manouni and M. Kbiri Alaoui, A result on elliptic systems with Neumann conditions via Ricceri's three critical points theorem, Nonlinear Anal. 71 (2009), no. 5-6, 2343-2348. https://doi.org/10.1016/j.na.2009.01.068
  9. J. R. Graef, S. Heidarkhani, and L. Kong, A critical points approach to multiplicity results for multi-point boundary value problems, Appl. Anal. 90 (2011), no. 12, 1909-1925. https://doi.org/10.1080/00036811.2010.534729
  10. S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters, Nonlinear Anal. 73 (2010), no. 2, 547-554. https://doi.org/10.1016/j.na.2010.03.051
  11. S. Heidarkhani and Y. Tian, Three solutions for a class of gradient Kirchhoff-type systems depending on two parameters, Dynam. Systems Appl. 20 (2011), no. 4, 551-562.
  12. C. Li and C.-L. Tang, Three solutions for a class of quasilinear elliptic systems involving the (p, q)-Laplacian, Nonlinear Anal. 69 (2008), no. 10, 3322-3329. https://doi.org/10.1016/j.na.2007.09.021
  13. C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Anal. 72 (2010), no. 3-4, 1339-1347. https://doi.org/10.1016/j.na.2009.08.011
  14. L. Li and C.-L. Tang, Existence of three solutions for (p, q)-biharmonic systems, Non-linear Anal. 73 (2010), no. 3, 796-805. https://doi.org/10.1016/j.na.2010.04.018
  15. J. Liu and X. Shi, Existence of three solutions for a class of quasilinear elliptic systems involving the $(p({\mathit{x}}), q({\mathit{x}}))$-Laplacian, Nonlinear Anal. 71 (2009), no. 1-2, 550-557. https://doi.org/10.1016/j.na.2008.10.094
  16. X.-L. Liu andW.-T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters, J. Math. Anal. Appl. 327 (2007), no. 1, 362-375. https://doi.org/10.1016/j.jmaa.2006.04.021
  17. A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal. 31 (1998), no. 7, 895-908. https://doi.org/10.1016/S0362-546X(97)00446-X
  18. B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems, Math. Comput. Modelling 32 (2000), no. 11-13, 1485-1494. https://doi.org/10.1016/S0895-7177(00)00220-X
  19. B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. 70 (2009), no. 9, 3084-3089. https://doi.org/10.1016/j.na.2008.04.010
  20. J. Simon, Regularite de la solution d'une equation non lineaire dans ${\mathbb{R}^N}$, In Journees d'Analyse Non Lineaire (Proc. Conf., Besancon, 1977), volume 665 of Lecture Notes in Math., pages 205-227, Springer, Berlin, 1978.
  21. J. Sun, H. Chen, J. J. Nieto, and M. Otero-Novoa, The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear Anal. 72 (2010), no. 12, 4575-4586. https://doi.org/10.1016/j.na.2010.02.034
  22. E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B, Nonlinear monotone operators. Translated from the German by the author and Leo F. Boron.Springer-Verlag, New York, 1990.
  23. L. Zhang and W. Ge, Solvability of a kind of Sturm-Liouville boundary value problems with impulses via variational methods, Acta Appl. Math. 110 (2010), no. 3, 1237-1248. https://doi.org/10.1007/s10440-009-9504-7

Cited by

  1. Existence of Multiple Solutions for a Quasilinear Biharmonic Equation vol.2014, 2014, https://doi.org/10.1155/2014/370494