Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms

Tetsutaro Shibata

doi:10.3934/cpaa.2018102

Article Contents

2018, Volume 17, Issue 5: 2139-2147. Doi: 10.3934/cpaa.2018102

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Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms

Tetsutaro Shibata^,

Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

* Corresponding author
* Corresponding author

Received: January 31, 2017

Revised: May 31, 2017

Published: April 2018

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Abstract

We consider the bifurcation problem

$-u''\left(t \right) = \lambda \left(u\left(t \right)+g\left(u\left(t \right) \right) \right),\;u\left(t \right)>0,\;\;t\in I: = \left(-1, 1 \right),\;\; u\left(\pm 1 \right) = 0, $

where $g(u) ∈ C^1(\mathbb{R}) $ is a periodic function with period 2π and $λ > 0$ is a bifurcation parameter. It is known that, under the appropriate conditions on $g$, $λ$ is parameterized by the maximum norm $α = \Vert u_λ\Vert_∞$ of the solution $u_λ$ associated with $λ$ and is written as $λ = λ(α)$. If $g(u)$ is periodic, then it is natural to expect that $λ(α)$ is also oscillatory for $α \gg 1$. We give a simple condition of $g(u)$, by which we can easily check whether $λ(α)$ is oscillatory and intersects the line $λ = π^2/4$ infinitely many times for $\alpha \gg 1$ or not.

Keywords:

global and local structure of bifurcation curves,
periodic nonlinear terms.

Mathematics Subject Classification: Primary: 34C23, 34F10; Secondary: 34B15.

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Figure 1. λ(g; α) with (OP)

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Figure 2. graph of sinx + $\epsilon\psi$(x)

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