Electron. J. Diff. Equ., Vol. 2016 (2016), No. 201, pp. 1-11.

Global and local behavior of the bifurcation diagrams for semilinear problems

Tetsutaro Shibata

We consider the nonlinear eigenvalue problem
 u''(t) + \lambda (u(t)^p - u(t)^q) = 0, \quad u(t) > 0,\quad -1<t<1,\cr
 u(1) = u(-1) = 0,
where $1 < p < q$ are constants and $\lambda > 0$ is a parameter. It is known in [13] that the bifurcation curve $\lambda(\alpha)$ consists of two branches, which are denoted by $\lambda_\pm(\alpha)$. Here, $\alpha = \| u_\lambda\|_\infty$. We establish the asymptotic behavior of the turning point $\alpha_p$ of $\lambda(\alpha)$, namely, the point which satisfies $d\lambda(\alpha_p)/d\alpha = 0$ as $p \to q$ and $p \to 1$. We also establish the asymptotic formulas for $\lambda_{+}(\alpha)$ and $\lambda_{-}(\alpha)$ as $\alpha \to 1$ and $\alpha \to 0$, respectively.

Submitted June 20, 2016. Published July 27, 2016.
Math Subject Classifications: 34F10.
Key Words: Asymptotic behavior; parabola-like bifurcation curves.

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Tetsutaro Shibata
Laboratory of Mathematics
Institute of Engineering
Hiroshima University
Higashi-Hiroshima, 739-8527, Japan
email: shibata@amath.hiroshima-u.ac.jp

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