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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:291

Article

Wang, SH ; Yeh, TS

关键词: bifurcation diagram;    positive solution;    exact multiplicity;    solution curve;    concave-convex nonlinearity;    time map;   

DOI  :  10.1016/j.jmaa.2003.10.021

来源: Elsevier

【 摘 要 】

We study the bifurcation diagrams of positive solutions of the two point boundary value problem {u(x)+f(lambda)(u(x))=0 -1 < x < 1 {u(-1) = u(1) = 0, where f(lambda)(u) =lambdag(u) + h(u), g, h is an element of C [0, infinity) boolean AND C-2 (0, infinity), and lambda > 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 (lambda, parallel touparallel to(infinity))-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 lambda > 0. More precisely, we prove the exact multiplicity of positive solutions. (C) 2003 Elsevier Inc. All. rights reserved.

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