Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 62, pp. 1-11.

Title: Asymptotic formulas for oscillatory bifurcation diagrams of
       semilinear ordinary differential equations

Author: Tetsutaro Shibata (Hiroshima Univ., Higashi-Hiroshima, Japan)

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 $p < 1$. The main tools of the proof are
 time-map argument and  stationary phase method. 


Submitted October 31, 2018. Published May 07, 2019.
Math Subject Classifications: 34C23, 34F10.
Key Words: Oscillatory bifurcation; time-map argument; stationary phase method.