Bifurcation theory for a class of second order differential equations

We consider the existence of positive solutions of the nonlinear two point boundary value problem u′′ + λf(u) = 0, u(−1) = u(1) = 0, where f(u) = u(u − a)(u− b)(u− c)(1−u), 0 < a < b < c < 1, as the parameter λ varies through positive values. Every solution u(x) is an even function, and when it exists, it is uniquely identified by α = u(0). We study how the number of solutions changes when the parameter varies, i.e. we will be focusing on the locations of bifurcation points. The authors P. Korman, Y. Li and T. Ouyang ( ”Computing the location and the direction of bifurcation”, Mathematical Research Letters, accepted ), prove that a necessary and sufficient condition for α to be a bifurcation point is G(α) ≡ F (α) ∫ α 0 f(α)− f(τ) [F (α)− F (τ)]3/2 dτ − 2 = 0, where F (α) = ∫ α 0 f(u) du. We will prove that G(α) has vertical asymptotes at α = b, α = 1 and at any point α ∈ (0, 1) for which ∫ α 0 f(u) du = 0. We will use the asymptotic behavior of G to estimate intervals where G(α) 6= 0, that is, intervals where there is no bifurcation point.