A complete classification of bifurcation diagrams of a Dirichlet problem with concave–convex nonlinearities

We study the bifurcation diagrams of positive solutions of the two point boundary value problem u (x) + fλ(u(x)) = 0, −10 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