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1997 Spectral asymptotics of nonlinear multiparameter Sturm-Liouville problems
Tetsutaro Shibata
Differential Integral Equations 10(4): 625-648 (1997).

Abstract

We consider the following nonlinear multiparameter Sturm-Liouville problem: $$ \begin{align} u''(x) + \sum_{k=1}^n \mu_kf_k(u(x)) &= \lambda g(u(x)), \,\, u(x) > 0, \,\, x \in I := (0,1), \\ u(0) &= u(1) = 0, \end{align} $$ where $\mu = (\mu_1, \mu_2, \cdots, \mu_n) \in R_+^n \,\, (R_+ := (0, \infty))$ and $\lambda \in R_+$ are parameters. By using Ljusternik-Schnirelman theory on general level set due to Zeidler, the variational eigenvalues $\lambda = \lambda(\mu, \alpha)$ are obtained. Here, $\alpha > 0$ is a parameter of general level sets. We shall establish an asymptotic formula of $\lambda(\mu, \alpha)$ as $\mu_1 \to \infty$.

Citation

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Tetsutaro Shibata. "Spectral asymptotics of nonlinear multiparameter Sturm-Liouville problems." Differential Integral Equations 10 (4) 625 - 648, 1997.

Information

Published: 1997
First available in Project Euclid: 1 May 2013

zbMATH: 0894.34016
MathSciNet: MR1741766

Subjects:
Primary: 34B15
Secondary: 34L30, 47J30, 58E05

Rights: Copyright © 1997 Khayyam Publishing, Inc.

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Vol.10 • No. 4 • 1997
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