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

where , are given constants satisfying , and is a parameter. It is known that under suitable conditions on , is parameterized by the maximum norm of the solution associated with and is a continuous function for . When , and , this equation has been introduced by Chen [4] as a model equation such that there exist infinitely many solutions near . We prove that is an oscillatory bifurcation curve as by showing the asymptotic formula for . It is found that the shapes of bifurcation curves depend on the condition or . 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