Boundary Value Problems
Volume 2010 (2010), Article ID 207649, 16 pages
Research Article

Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function

1Department of Mathematics, Shanghai Institute of Technology, Shanghai 200235, China
2School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China

Received 6 March 2010; Revised 18 July 2010; Accepted 11 August 2010

Academic Editor: Raul F. Manasevich

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.


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.