Classification and Evolution of Bifurcation Curves for a Dirichlet-Neumann Boundary Value Problem and its Application

Abstract

We study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Dirichlet-Neumann boundary value problemwhere 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 c 1 > 0, such that, on the (λ, u ∞ )-plane, (i) when 0 < c < c 1 , the bifurcation curve is S-shaped;(ii) when c ≥ c 1 , the bifurcation curve is ⊂-shaped. Our results can be applied to the one-dimensional perturbed Gelfand equation with f (u) = exp au a+u for a ≥ 4.37.