International Journal of Differential Equations

Volume 2015 (2015), Article ID 138629, 11 pages

http://dx.doi.org/10.1155/2015/138629

## 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 .

#### 1. Introduction

We consider the following nonlinear eigenvalue problem: where is a cubic-like nonlinear term and is a parameter. We assume the following conditions (A.1)–(A.3), which have been introduced in [1].(A.1) is a -function on and has three positive roots at , and (A.2)There exists a constant such that (A.3)Let satisfy . Then The typical example of which satisfies (A.1)–(A.3) is with and the area of the negative hump of is nearly equal to that of the positive hump. For example, if we choose appropriately, then satisfies (A.1)–(A.3).

Nonlinear elliptic eigenvalue problems have been studied by many authors. We refer to [2–8] and the references therein. Among other things, (1)–(3) have been investigated by many authors. We refer to [1, 9–12] and the references therein. In particular, the following basic properties of the structure of bifurcation diagram for (1)–(3) have been proved in [1, Theorem 3.1].

Theorem 1 (see [1, Theorem 3.1]). *Assume (A.1)–(A.3). Then there exists a critical such that (1)–(3) have exactly one positive solution for , exactly two positive solutions for , and exactly three positive solutions for . Furthermore, all solutions lie on two smooth solution curves, and is parameterized by as . One of the curves, referred to as the lower curve , satisfies and increases in , and
**
where is a solution of (1)–(3) corresponding to . (Note that is equivalent to .) The upper curve, which consists of two branches (), is a parabola-like curve with exactly one turn to the right at . Furthermore,
**
where is a solution of (1)–(3) corresponding to () and is a constant which satisfies
*

*(See Figure 1 for the bifurcation diagram.)*