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

Abstract

We study the bifurcation diagrams of positive solutions of the multiparameter Dirichlet problem { u ″ ( x ) + f λ , μ ( u ( x ) ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , where f λ , μ ( u ) = g ( u , λ ) + h ( u , μ ) , λ > λ 0 and μ > μ 0 are two bifurcation parameters, λ 0 and μ 0 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 > μ 0 (resp. λ > λ 0 ), 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 > λ 0 (resp. μ > μ 0 ). 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.