Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity

Abstract

We study the global bifurcation and exact multiplicity of positive solutions of{u″(x)+λfε(u)=0,−1<x<1,u(−1)=u(1)=0,fε(u)=−εu3+σu2−κu+ρ, where λ,ε>0 are two bifurcation parameters, and σ,ρ>0, 0<κ⩽σρ are constants. We prove the global bifurcation of bifurcation curves for varying ε>0. More precisely, there exists ε˜>0 such that, on the (λ,‖u‖∞)-plane, the bifurcation curve is S-shaped for 0<ε<ε˜ and is monotone increasing for ε⩾ε˜. Thus we are able to determine the exact number of positive solutions by the values of ε and λ. Our results extend those of Hung and Wang (K.-C. Hung, S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., in press) from κ⩽0 to κ⩽σρ.