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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:349

Article

Wang, Shin-Hwa¹ 

[1] Natl Tsing Hua Univ, Dept Math, Hsinchu 300, Taiwan

关键词: Bifurcation diagram;    Positive solution;    Multiparameter problem;    Concave-convex nonlinearity;    Exact multiplicity;    Time map;   

DOI  :  10.1016/j.jmaa.2008.08.020

来源: Elsevier

【 摘 要 】

We study the bifurcation diagrams of positive solutions of the multiparameter Dirichlet problem {u ''(x) + f(lambda.mu) (u(x)) = 0. -1 < x < 1. u(-1) = u(1) = 0. where f(lambda.mu) (u) = g(u,lambda) + h(u,mu), lambda > lambda(0) and mu > mu(0) are two bifurcation parameters,) lambda(0) and Ito 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 mu > mu(0) (resp. lambda > lambda(0)). we give a classification of totally eight qualitatively different bifurcation diagrams. We prove that, on the (lambda. parallel to u parallel to(infinity))-plane (resp. (mu, parallel to u parallel 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>lambda(0) (resp. mu > mu(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 It (resp.;) varies.

【 授权许可】

Free