Abstract
We study the classification and evolution of bifurcation curves of positive solutions for the Dirichlet-Neumann boundary value problem u''(x)+λf(u)=0, 0<x<1, u(0)=0, u'(1)=-c<0, where λ>0 is a bifurcation parameter and c>0 is an evolution parameter. We mainly prove that, under some suitable assumptions on f, there exists c₁>0, such that, on the (λ,‖u‖∞)-plane, (i) when 0<c<c₁, the bifurcation curve is S-shaped and the problem has at least three positive solutions for some range of positive λ; (ii) when c≥c₁, the bifurcation curve is ⊂-shaped and the problem has at least two positive solutions for some range of positive λ. Our results can be applied to the one-dimensional perturbed Gelfand equation with f(u)=exp((au)/(a+u)) for a≥4.37.