A complete classification of bifurcation diagrams of classes of multiparameter p-Laplacian boundary value problems

We study the bifurcation diagrams of positive solutions of the multiparameter p-Laplacian problem { (phi(p)(u'(x)))' + f(lambda.mu) (u(x)) = 0, -1 < x < 1. u(-1) = u(1) = 0, where p > 1, phi(p)(y) = vertical bar y broken vertical bar(p-2)y, (phi(p)(u'))' is the one-dimensional p-Laplacian, f(lambda.mu)(u) = g(u, lambda) + h(u, mu), and lambda > lambda(0) and mu > mu(0) are two bifurcation parameters, lambda(0) and mu(0) are two given real numbers. Assuming that functions g and h satisfy hypotheses (H1)-(H3) and (H4a) (resp. (H1)-(H3) and (H4b)), for fixed mu > mu(0) (resp. lambda > lambda(0)), we give a classification of totally eight qualitatively different bifurcation diagrams. We prove that, on the (lambda. parallel to u parallel to(infinity))-plane (resp. (mu. parallel to u parallel to(infinity))-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 right. Hence the problem has at most two positive solutions for each lambda > lambda(0) (resp. mu > mu(0)). More precisely, we prove the exact multiplicity of positive solutions. In addition, for all p > 1, we give interesting examples which show complete evolution of bifurcation diagrams as mu (resp. lambda) varies. (C) 2008 Elsevier Inc. All rights reserved.