Differential and Integral Equations

Spectral asymptotics of nonlinear multiparameter Sturm-Liouville problems

Tetsutaro Shibata

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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 $\mu_1 \to \infty$.

Article information

Differential Integral Equations, Volume 10, Number 4 (1997), 625-648.

First available in Project Euclid: 1 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B15: Nonlinear boundary value problems
Secondary: 34L30: Nonlinear ordinary differential operators 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Shibata, Tetsutaro. Spectral asymptotics of nonlinear multiparameter Sturm-Liouville problems. Differential Integral Equations 10 (1997), no. 4, 625--648. https://projecteuclid.org/euclid.die/1367438635

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