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