Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 37, pp. 1-16.
Title: Asymptotic shape of solutions to nonlinear eigenvalue problems
Author: Tetsutaro Shibata (Hiroshima Univ., Japan)
Abstract:
We consider the nonlinear eigenvalue problem
$$
-u''(t) = f(\lambda, u(t)), \quad u > 0, \quad u(0) = u(1) = 0,
$$
where $\lambda > 0$ is a parameter. It is known that under some
conditions on $f(\lambda, u)$, the shape of the solutions associated with
$\lambda$ is almost `box' when $\lambda \gg 1$.
The purpose of this paper is to study precisely the asymptotic shape of
the solutions as $\lambda \to \infty$ from a standpoint of $L^1$-framework.
To do this, we establish the asymptotic formulas for
$L^1$-norm of the solutions as $\lambda \to \infty$.
Submitted January 11, 2005. Published March 29, 2005.
Math Subject Classifications: 34B15.
Key Words: Asymptotic formula; $L^1$-norm; simple pendulum; logistic equation