Electron. J. Differential Equations, Vol. 2019 (2019), No. 62, pp. 1-11.

Asymptotic formulas for oscillatory bifurcation diagrams of semilinear ordinary differential equations

Tetsutaro Shibata

Abstract:
We study the nonlinear eigenvalue problem
$$ -u''(t) = \lambda \left(u(t)^p + g(u(t))\right), \quad u(t) > 0, \quad t \in (-1,1), \; u(\pm 1) = 0, $$
where $g(u) = h(u)\sin (u^r)$, $p, r$ are given constants satisfying $p \ge 0$, $0 < r \le 1$ and $\lambda > 0$ is a parameter. It is known that under suitable conditions on $h$, $\lambda$ is parameterized by the maximum norm $\alpha = \| u_\alpha\|_\infty$ of the solution $u_\lambda$ associated with $\lambda$ and $\lambda = \lambda(\alpha)$ is a continuous function for $\alpha > 0$. When $p = 1$, $h(u) \equiv 1$ and $r = 1/2$, this equation has been introduced by Chen [4] as a model equation such that there exist infinitely many solutions near $\lambda = \pi^2/4$. We prove that $\lambda(\alpha)$ is an oscillatory bifurcation curve as $\alpha \to \infty$ by showing the asymptotic formula for $\lambda(\alpha)$. It is found that the shapes of bifurcation curves depend on the condition $p > 1$ or $p < 1$. time-map argument and stationary phase method.

Submitted October 31, 2018. Published May 7, 2019.
Math Subject Classifications: 34C23, 34F10.
Key Words: Oscillatory bifurcation; time-map argument; stationary phase method.

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Tetsutaro Shibata
Laboratory of Mathematics
Graduate School of Engineering
Hiroshima University
Higashi-Hiroshima, 739-8527, Japan
email: tshibata@hiroshima-u.ac.jp

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