Electron. J. Differential Equations,
Vol. 2019 (2019), No. 62, pp. 111.
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
.
timemap argument and stationary phase method.
Submitted October 31, 2018. Published May 7, 2019.
Math Subject Classifications: 34C23, 34F10.
Key Words: Oscillatory bifurcation; timemap argument; stationary phase method.
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Tetsutaro Shibata
Laboratory of Mathematics
Graduate School of Engineering
Hiroshima University
HigashiHiroshima, 7398527, Japan
email: tshibata@hiroshimau.ac.jp

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