Boundary Value Problems
Volume 2008 (2008), Article ID 728603, 8 pages
doi:10.1155/2008/728603

Multiple positive solutions for singular quasilinear multipoint BVPs with the first-order derivative

Weihua Jiang1 , Bin Wang2 and Yanping Guo1

1College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018 Hebei, China
 2Department of Basic Courses, Hebei Professional and Technological College of Chemical and Pharmaceutical Engineering, Shijiazhuang, 050031 Hebei, China

Abstract

The existence of at least three positive solutions for differential equation (ϕp(u′(t)))′+g(t)f(t,u(t),u′(t))=0, under one of the following boundary conditions: u(0)=∑i=1m−2aiu(ξi), φp(u′(1))=∑i=1m−2biφp(u′(ξi)) or φp(u′(0))=∑i=1m−2aiφp(u′(ξi)), u(1)=∑i=1m−2biu(ξi) is obtained by using the H. Amann fixed point theorem, where φp(s)=|s|p−2s, p>1, 0<ξ1<ξ2<⋯<ξm−2<1, ai>0, bi>0, 0<∑i=1m−2ai<1, 0<∑i=1m−2bi<1. The interesting thing is that g(t) may be singular at any point of [0,1] and f may be noncontinuous.