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 $\mu_1 \to \infty$.