Second order spectral estimates and symmetry breaking for rotating wave solutions

Abstract

We consider rotating wave solutions of the nonlinear wave equation \[ \left\{ \begin{aligned} \partial_{t}^2 v - Δv + m v & = |v|^{p-2} v \quad && \text{in $\mathbb{R} \times \textbf{B}$} \\ v & = 0 && \text{on $\mathbb{R} \times \partial \textbf{B}$} \end{aligned} \right. \] for $21$. We find that the structure of the spectrum of $L_α$ strongly depends on the quantity \[ σ= \fracπ{\sqrt{α^2- 1} - \arccos \frac{1}α} > 0 . \] By giving precise estimates for certain sequences of Bessel function zeros, we can classify the spectrum for all $α>1$ such that $σ$ is rational and further find that the existence of accumulation points explicitly depends on arithmetic properties of $σ$. Using these characterizations, we deduce existence and symmetry breaking results for ground state solutions of the reduced equation, extending known results.