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{(φp(u′(x)))′+fλ,μ(u(x))=0,−11p>1, φp(y)=|y|p−2yφp(y)=|y|p−2y, (φp(u′))′(φp(u′))′ is the one-dimensional p  -Laplacian, fλ,μ(u)=g(u,λ)+h(u,μ)fλ,μ(u)=g(u,λ)+h(u,μ), and λ>λ0λ>λ0 and μ>μ0μ>μ0 are two bifurcation parameters, λ0λ0 and μ0μ0 are two given real numbers. Assuming that functions g and h   satisfy hypotheses (H1)–(H3) and (H4a) (resp. (H1)–(H3) and (H4b)), for fixed μ>μ0μ>μ0 (resp. λ>λ0λ>λ0), we give a classification of totally eight   qualitatively different bifurcation diagrams. We prove that, on the (λ,‖u‖∞)(λ,‖u‖∞)-plane (resp. (μ,‖u‖∞)(μ,‖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>λ0λ>λ0 (resp. μ>μ0μ>μ0). More precisely, we prove the exact multiplicity of positive solutions. In addition, for all p>1p>1, we give interesting examples which show complete evolution of bifurcation diagrams as μ (resp. λ) varies.