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

Title & Authors
EXISTENCE OF THREE SOLUTIONS FOR A CLASS OF NAVIER QUASILINEAR ELLIPTIC SYSTEMS INVOLVING THE (p1, …, pn)-BIHARMONIC
Li, Lin;

Abstract
In this paper, we establish the existence of at least three solutions to a Navier boundary problem involving the ($\small{p_1}$, $\small{{\cdots}}$, $\small{p_n}$)-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
($\small{p_1}$, $\small{{\cdots}}$, $\small{p_n}$)-biharmonic;Navier condition;multiple solutions;three critical points theorem;
Language
English
Cited by
1.
Existence of Multiple Solutions for a Quasilinear Biharmonic Equation, International Scholarly Research Notices, 2014, 2014, 1
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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

17.
A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal. 31 (1998), no. 7, 895-908.

18.
B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems, Math. Comput. Modelling 32 (2000), no. 11-13, 1485-1494.

19.
B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. 70 (2009), no. 9, 3084-3089.

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.

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.