On the spectrum of a mixed-type operator with applications to rotating wave solutions

Abstract

We study rotating wave solutions of the nonlinear wave equation $$ \left\{ \begin{aligned} \partial_{t}^2 v - \Delta v + m v & = |v|^{p-2} v \quad && \text{in $\mathbb{R} \times \mathbf{B}$} \\ v & = 0 && \text{on $\mathbb{R} \times \partial \mathbf{B}$} \end{aligned} \right. $$ where $2<p<\infty$, $m \in \mathbb{R}$ and $\mathbf{B} \subset \mathbb{R}^2$ denotes the unit disk. If the angular velocity $\alpha$ of the rotation is larger than $1$, this leads to a semilinear boundary value problem on $\mathbf{B}$ involving a mixed-type operator, whose spectrum is related to the zeros of Bessel functions and could generally be badly behaved. Based on new estimates for these zeros, we find values of $\alpha$ such that the spectrum only consists of eigenvalues with finite multiplicity and has no accumulation point. Combined with suitable spectral estimates, this allows us to formulate an appropriate indefinite variational setting and find ground state solutions of the reduced equation for $p \in (2,4)$. Using a minimax characterization of the ground state energy, we ultimately show that these ground states are nonradial and thus yield nontrivial rotating waves, provided $m$ is sufficiently large.