# Global and local structures of oscillatory bifurcation curves

### Tetsutaro Shibata

Hiroshima University, Higashi-Hiroshima, Japan

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## Abstract

We consider the nonlinear eigenvalue problem \begin{align*} -u''(t) &= \lambda (u(t) + g(u(t))), \: u(t) > 0, \: t \in I := (-1,1),\\[1mm] u(\pm 1) &= 0, \end{align*} where $g(u) = u^p\\sin(u^q)$ ($0 \\le p < 1$, $0 < q \\le 1$) and $\\lambda > 0$ is a bifurcation parameter. It is known that, in this case, $\\lambda$ is parameterized by the maximum norm $\\alpha = \\Vert u\_\\lambda\\Vert\_\\infty$ of the solution $u\_\\lambda$ associated with $\\lambda$ and is written as $\\lambda = \\lambda(\\alpha)$. We show that the bifurcation curve $\\lambda(\\alpha)$ intersects the line $\\lambda = \\pi^2/4$ infinitely many times by establishing the precise asymptotic formula for $\\lambda(\\alpha)$ as $\\alpha \\to \\infty$ and $\\alpha \\to 0$. We find that, according to the relationship between $p$ and $q$, there exist three types of bifurcation curves.

## Cite this article

Tetsutaro Shibata, Global and local structures of oscillatory bifurcation curves. J. Spectr. Theory 9 (2019), no. 3, pp. 991–1003

DOI 10.4171/JST/269