| Abstract: |
| We consider the bifurcation problem
$-u``(t) = \lambda \left(u(t)
+ g(u(t))\right)$, $u(t) > 0$, \enskip $t \in I := (-1,1)$,
\enskip
$u(-1) = u(1) = 0$,
where $g(u)$ is a nonlinear term
and $\lambda > 0$ is a bifurcation parameter.
It is known that under some suitable conditions on $g(u)$,
$\lambda$ is
parameterized by
the maximum norm $\alpha = \Vert u_\lambda\Vert_\infty$ of the solution
$u_\lambda$
associated with $\lambda$
and is written as $\lambda = \lambda(g,\alpha)$.
We show that if $g(u)$ satisfies some oscillatory conditions, then
the bifurcation diagram of $\lambda(g,\alpha)$ intersects the line
$\lambda = \pi^2/4$ infinitely
many times by establishing the precise asymptotic formulas for
$\lambda(g,\alpha)$ as $\alpha \to \infty$.
We also establish the precise asymptotic formulas for
$\lambda(g,\alpha)$ as $\alpha \to 0$. |
|