Electron. J. Diff. Eqns., Vol. 2005(2005), No. 37, pp. 1-16.

Asymptotic shape of solutions to nonlinear eigenvalue problems

Tetsutaro Shibata

Abstract:
We consider the nonlinear eigenvalue problem
$$ -u''(t) = f(\lambda, u(t)), \quad u \mbox{greater than} 0, \quad u(0) = u(1) = 0, $$
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

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Tetsutaro Shibata
Department of Applied Mathematics
Graduate School of Engineering
Hiroshima University
Higashi-Hiroshima, 739-8527, Japan
email: shibata@amath.hiroshima-u.ac.jp

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