Abstract
We study the global bifurcation and exact multiplicity of positive solutions of, 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, {norm of matrix}u{norm of matrix} <sub>∞</sub> )-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. © 2012 Elsevier Inc.