Multiplicity Results for a Kirchho-type Doubly Eigenvalue Boundary Value Problem

Volume 3, Issue 1, pp 11--20
• 2011 Views

Authors

S. Heidarkhani - Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran G. A. Afrouzi - Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran, 47416-1467 Babolsar, Iran

Abstract

This paper is concerned with the existence of at least three weak solutions to a class of Kirchhoff-type doubly eigenvalue boundary value problem. The technical approach is mainly based on a very recent three critical points theorem due to B. Ricceri [On a three critical points theorem revisited, Nonlinear Anal., 70 (2009) 3084-3089.]

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ISRP Style

S. Heidarkhani, G. A. Afrouzi, Multiplicity Results for a Kirchho-type Doubly Eigenvalue Boundary Value Problem, Journal of Mathematics and Computer Science, 3 (2011), no. 1, 11--20

AMA Style

Heidarkhani S., Afrouzi G. A., Multiplicity Results for a Kirchho-type Doubly Eigenvalue Boundary Value Problem. J Math Comput SCI-JM. (2011); 3(1):11--20

Chicago/Turabian Style

Heidarkhani, S., Afrouzi, G. A.. "Multiplicity Results for a Kirchho-type Doubly Eigenvalue Boundary Value Problem." Journal of Mathematics and Computer Science, 3, no. 1 (2011): 11--20

Keywords

• Kirchhoff-type problem
• Critical point
• Three solutions
• Variational methods.

•  35K20
•  49R05

References

• [1] G. A. Afrouzi, S. Heidarkhani, Three solutions for a quasilinear boundary value problem, Nonlinear Anal., 69 (2008), 3330--3336

• [2] C. O. Alves, F. Corrêa, T. F. Ma, Positive solutions for a quasilinear elliptic equations of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85--93

• [3] G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal., 54 (2003), 651--665

• [4] G. Bonannao, Existence of three solutions for a two point boundary value problem, Appl. Math. Lett., 13 (2000), 53--57

• [5] P. Candito, Existence of three solutions for a nonautonomous two point boundary value problem, J. Math. Anal. Appl., 252 (2000), 532--537

• [6] M. Chipot, B. Lovat, Some remarks on non local elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619--4627

• [7] X. He, W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407--1414

• [8] S. Heidarkhani, D. Motreanu, Multiplicity results for a two-point boundary value problem, Panamer. Math. J., 19 (2009), 69--78

• [9] R. Livrea, Existence of three solutions for a quasilinear two point boundary value problem, Arch. Math., 79 (2002), 288--298

• [10] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1957--1977

• [11] S. A. Marano, D. Motreanu, On a three critical points theorem for non- differentiable functions and applications nonlinear boundary value problems, Nonlinear Anal., 48 (2002), 37--52

• [12] A. Mao, Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type prob- lems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275--1287

• [13] K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246--255

• [14] B. Ricceri , A further three critical points theorem, Nonlinear Anal., 71 (2009), 4151--4157

• [15] B. Ricceri, A three critical points theorem revisited, Nonlinear Anal., 70 (2009), 3084--3089

• [16] B. Ricceri, On a three critical points theorem, Arch. Math., 75 (2000), 220--226

• [17] B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optimization, 46 (2010), 543--549

• [18] E. Zeidler, Nonlinear functional analysis and its applications, Springer-Verlag, New York (1985)

• [19] Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456--463