Electron. J. Diff. Equ., Vol. 2011 (2011), No. 91, pp. 1-11.

Existence of three solutions for a Kirchhoff-type boundary-value problem

Shapour Heidarkhani, Ghasem Alizadeh Afrouzi, Donal O'Regan

In this note, we establish the existence of two intervals of positive real parameters $\lambda$ for which the boundary-value problem of Kirchhoff-type
 -K\big(\int_{a}^b |u'(x)|^2dx\big)u''=\lambda f(x,u),\cr
admits three weak solutions whose norms are uniformly bounded with respect to $\lambda$ belonging to one of the two intervals. Our main tool is a three critical point theorem by Bonanno.

Submitted April 26, 2011. Published July 6, 2011.
Math Subject Classifications: 35J20, 35J25, 35J60.
Key Words: Kirchhoff-type problem; multiple solutions; critical point.

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Shapour Heidarkhani
Department of Mathematics
Faculty of Sciences, Razi University
67149 Kermanshah, Iran
email: s.heidarkhani@razi.ac.ir
Ghasem Alizadeh Afrouzi
Department of Mathematics, Faculty of Basic Sciences
University of Mazandaran, 47416-1467 Babolsar, Iran
email: afrouzi@umz.ac.ir
Donal O'Regan
Department of Mathematics
National University of Ireland
Galway, Ireland
email: donal.oregan@nuigalway.ie

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