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

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 $\lambda(\mu)$.