Copyright © 2010 Yulian An and Hua Luo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An exact multiplicity result of positive solutions for the boundary value problems u′′+λa(t)f(u)=0, t∈(0,1), u′(0)=0, u(1)=0 is achieved, where λ is a positive parameter. Here the function f:[0,∞)→[0,∞) is C2 and satisfies f(0)=f(s)=0, f(u)>0 for u∈(0,s)∪(s,∞) for some s∈(0,∞). Moreover, f is asymptotically linear and f″ can change sign only once. The weight function a:[0,1]→(0,∞) is C2 and satisfies a′(t)<0, 3(a′(t))2<2a(t)a′′(t) for t∈[0,1]. Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for λ lying in various intervals in R. Moreover, we indicate how to extend the result to the general case.