Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 91, pp. 1-11.

Title: Existence of three solutions for a Kirchhoff-type
       boundary-value problem
       
Authors: Shapour Heidarkhani (Razi Univ., Kermanshah, Iran)
         Ghasem Alizadeh Afrouzi (Univ. of Mazandaran, Babolsar, Iran)
	 Donal  O'Regan (National Univ. of Ireland, Galway, Ireland)

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 06, 2011.
Math Subject Classifications: 35J20, 35J25, 35J60.
Key Words: Kirchhoff-type problem; multiple solutions; critical point.