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International Journal of Differential Equations
Volume 2015, Article ID 138629, 11 pages
http://dx.doi.org/10.1155/2015/138629
Research Article

Asymptotic Behavior of the Bifurcation Diagrams for Semilinear Problems with Application to Inverse Bifurcation Problems

Laboratory of Mathematics, Institute of Engineering, Hiroshima University, Higashihiroshima 739-8527, Japan

Received 22 October 2014; Accepted 15 December 2014

Academic Editor: Nasser-Eddine Tatar

Copyright © 2015 Tetsutaro Shibata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the nonlinear eigenvalue problem ,  ,  ,  , where is a cubic-like nonlinear term and is a parameter. It is known by Korman et al. (2005) that, under the suitable conditions on , there exist exactly three bifurcation branches (), and these curves are parameterized by the maximum norm of the solution corresponding to . In this paper, we establish the precise global structures for (), which can be applied to the inverse bifurcation problems. The precise local structures for () are also discussed. Furthermore, we establish the asymptotic shape of the spike layer solution , which corresponds to , as .