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Article
Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities
Journal of Differential Equations (2014)
  • Philip Korman
  • 8186772957 Yi Li
Abstract
In this paper the authors study exact multiplicity of positive solutions of the two-point boundary value problem
u+λf(u)=0, −1<x<1,u(−1)=u(1)=0,λ>0.(1)
They also study the linearized problem corresponding to this problem:
w+λf(u)w=0, −1<x<1,w(−1)=w(1)=0.
Lemma 2.1 gives the connection between critical points (λ,u) of problem (1) and a nontrivial solution of the corresponding problem for w. Lemma 2.2 and Theorem 2.1 are results from previous work. The main result is Theorem 2.2, which has two parts (when f is convex-concave and when it is concave-convex) and gives conditions under which the global solution curve admits at most two turns at critical points. Theorem 2.3, Theorem 2.4, Theorem 2.5 and Theorem 2.6 are corollaries of Theorem 2.2.
   This paper is correct and has new results in this theory.
Keywords
  • Exact multiplicity of positive solutions,
  • S-shaped and reversed S-shaped bifurcation
Publication Date
November 15, 2014
DOI
https://doi.org/10.1016/j.jde.2014.07.007
Citation Information
Philip Korman and Yi Li. "Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities" Journal of Differential Equations (2014)
Available at: http://works.bepress.com/yi_li/104/