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

Abstract:
In this note, we establish the existence of two intervals of positive real parameters $\lambda$ for which the boundary-value problem of Kirchhoff-type
$$\displaylines{ -K\big(\int_{a}^b |u'(x)|^2dx\big)u''=\lambda f(x,u),\cr u(a)=u(b)=0 }$$
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.

Show me the PDF file (241 KB), TEX file, and other files for this article.

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

Return to the EJDE web page