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Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity. (English) Zbl 1245.34048

The authors investigate the global bifurcation diagram and the exact multiplicity of positive solutions for the problem

u '' (x)+λf ε (u)=0,-1<x<1,u(-1)=u(1)=0,f ε (u)=-εu 3 +σu 2 -κu+ρ,

where λ,ε>0 are two bifurcation parameters, and σ,ρ>0, 0<κσρ are constants. They prove that 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 εε ˜.

34C23Bifurcation (ODE)
34B18Positive solutions of nonlinear boundary value problems for ODE
34B08Parameter dependent boundary value problems for ODE
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