A complete classification of bifurcation diagrams of classes of a multiparameter Dirichlet problem with concave-convex nonlinearities

AbstractWe study the bifurcation diagrams of positive solutions of the multiparameter Dirichlet problem{u″(x)+fλ,μ(u(x))=0,−1<x<1,u(−1)=u(1)=0, where fλ,μ(u)=g(u,λ)+h(u,μ), λ>λ0 and μ>μ0 are two bifurcation parameters, λ0 and μ0 are two given real numbers. Assuming that functions g and h satisfy hypotheses (H1)–(H3) and (H4)(a) (resp. (H1)–(H3) and (H4)(b)), for fixed μ>μ0 (resp. λ>λ0), we give a classification of totally eight qualitatively different bifurcation diagrams. We prove that, on the (λ,‖u‖∞)-plane (resp. (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 left. Hence the problem has at most two positive solutions for each λ>λ0 (resp. μ>μ0). More precisely, we prove the exact multiplicity of positive solutions. In addition, we give interesting examples which show complete evolution of bifurcation diagrams as μ (resp. λ) varies