Abstract
We study the bifurcation diagrams of positive solutions of the p-Laplacian Dirichlet problem (φp(u′(x)))′+fλ(u(x))=0,-1<x<1, u(-1)=u(1)=0,where fλ(u)=λg(u)+h(u), g,h∈C[0,∞) ∩C2(0,∞), p>1,φp(y)=yp-2y,(φp(u′))′ is the one-dimensional p-Laplacian, 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 right. Hence the problem has at most two positive solutions for each λ>0. More precisely, we prove the exact multiplicity of positive solutions. In addition, for p=2, we give interesting examples which show the evolution phenomena of bifurcation diagrams. © 2005 Elsevier Ltd. All rights reserved.