## Differential and Integral Equations

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

### Spectral asymptotics of nonlinear multiparameter Sturm-Liouville problems

#### 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 $\mu_1 \to \infty$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 1 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367438635

**Mathematical Reviews number (MathSciNet)**

MR1741766

**Zentralblatt MATH identifier**

0894.34016

**Subjects**

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.)

#### Citation

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