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

AbstractWe 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