Asymptotic expansion of the variational eigencurve for two-parameter simple pendulum-type equations (Q1879774)

This paper is concerned with the study of the nonlinear two-parameter simple pendulum-type equation \[ -u''(t)+\mu u^{p} = \lambda \sin u(t), \quad u(t)>0, \quad t \in I=(-T,T), \quad u(\pm T)=0, \] where \(p>1\) is a constant and \(\mu, \lambda >0\) are parameters. The author shows the existence of a variational eigencurve \(\lambda(\mu)\), which has sub-linear growth as \(\mu \to \infty\), by adopting an appropriate variational framework. Also, the author establishes the exact second term of \(\lambda( \mu)\) as \(\mu \to \infty\).