## Differential and Integral Equations

### Critical exponents of the asymptotic formulas for two-parameter variational eigencurves

Tetsutaro Shibata

#### Abstract

We consider the two-parameter nonlinear eigenvalue problem $$-u''(t) = \mu u(t) - \lambda(u(t) + u(t)^p), \ u(t) > 0, t \in I := (0, 1), \ \ u(0) = u(1) = 0,$$ where $p > 1$ is a constant and $\mu, \lambda > 0$ are parameters. We establish the precise asymptotic formulas for the variational eigencurve $\lambda = \lambda(\mu)$, which is defined on a general level set, as $\mu \to \infty$. Especially, we obtain new critical exponents $p = 7/5$, $p = 5/3$, $p = 2$, and $p = 5$ from the viewpoint of the asymptotics of the eigencurve $\lambda(\mu)$.

#### Article information

Source
Differential Integral Equations, Volume 25, Number 9/10 (2012), 899-914.

Dates
First available in Project Euclid: 20 December 2012

https://projecteuclid.org/euclid.die/1356012374

Mathematical Reviews number (MathSciNet)
MR2985686

Zentralblatt MATH identifier
1274.34035

Subjects
Primary: 34B15: Nonlinear boundary value problems

#### Citation

Shibata, Tetsutaro. Critical exponents of the asymptotic formulas for two-parameter variational eigencurves. Differential Integral Equations 25 (2012), no. 9/10, 899--914. https://projecteuclid.org/euclid.die/1356012374