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 problem {u ''(x) + lambda f (u) = 0, 0 < x < 1, u(0) = 0, u' (1) = -c < 0, where lambda > 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 (lambda,parallel to u parallel to(infinity))-plane, (i) when 0 < c < c(1), the bifurcation curve is S-shaped; (ii) when c >= c(1), the bifurcation curve is subset of-shaped. Our results can be applied to the one-dimensional perturbed Gelfand equation with f(u) = exp (au/a+u) for a >= 4.37.