Abstract
We study the bifurcation diagrams of positive solutions of the two point boundary value problem {u″(x) + f <sub>λ</sub> (u(x)) = 0, -1 < x < 1, {u(- 1) = u(1) = 0, where f <sub>λ</sub> (u) = λg(u) + h(u), g, h ∈ C[0, ∞) ∩ C <sup>2</sup> (0, ∞), and λ > 0 is a bifurcation parameter. We assume that functions g and h satisfy hypotheses (H1)-(H3). Under hypotheses (H1)-(H3), we give a complete classification of bifurcation diagrams, and we prove that, on the (λ, ∥u∥ ∞)-plane, each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the left. Hence the problem has at most two positive solutions for each λ > 0. More precisely, we prove the exact multiplicity of positive solutions. © 2003 Elsevier Inc. All rights reserved.