JournalsjstVol. 9, No. 3pp. 991–1003

Global and local structures of oscillatory bifurcation curves

  • Tetsutaro Shibata

    Hiroshima University, Higashi-Hiroshima, Japan
Global and local structures of oscillatory bifurcation curves cover

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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)=upsin(uq)g(u) = u^p\\sin(u^q) (0lep<10 \\le p < 1, 0<qle10 < q \\le 1) and lambda>0\\lambda > 0 is a bifurcation parameter. It is known that, in this case, lambda\\lambda is parameterized by the maximum norm alpha=Vertu_lambdaVert_infty\\alpha = \\Vert u\_\\lambda\\Vert\_\\infty of the solution u_lambdau\_\\lambda associated with lambda\\lambda and is written as lambda=lambda(alpha)\\lambda = \\lambda(\\alpha). We show that the bifurcation curve lambda(alpha)\\lambda(\\alpha) intersects the line lambda=pi2/4\\lambda = \\pi^2/4 infinitely many times by establishing the precise asymptotic formula for lambda(alpha)\\lambda(\\alpha) as alphatoinfty\\alpha \\to \\infty and alphato0\\alpha \\to 0. We find that, according to the relationship between pp and qq, 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