A complete classification of bifurcation diagrams of a p-Laplacian Dirichlet problem

We study the bifurcation diagrams of positive solutions of the p-Laplacian Dirichlet problem (ϕp(u′(x)))′+fλ(u(x))=0,-11,ϕp(y)=|y|p-2y,(ϕp(u′))′p>1,ϕp(y)=|y|p-2y,(ϕp(u′))′ is the one-dimensional p  -Laplacian, and λ>0λ>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∥∞)(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>0λ>0. More precisely, we prove the exact multiplicity of positive solutions. In addition, for p=2p=2, we give interesting examples which show the evolution phenomena of bifurcation diagrams.