Second Order Spectral Estimates and Symmetry Breaking for Rotating Wave Solutions

Abstract

We consider rotating wave solutions of the nonlinear wave equation \[ \left\{ \begin{aligned} \partial_{t}^2 v - Δv + m v & = |v|^{p-2} v \quad && \text{in $\mathbb{R}\times \text{B}$} \\ v & = 0 && \text{on $\mathbb{R}\times \partial \text{B}$} \end{aligned} \right. \] for $2<p<\infty$, $m\in \mathbb{R}$ on the unit disk $\text{B}\subset {\mathbb{R}}^2$. This leads to the study of a reduced equation involving the elliptic-hyperbolic operator $L_{\alpha }=-\Delta +{\alpha }^2{\partial }_{\theta }^2$ with $\alpha >1$. We find that the structure of the spectrum of $L_{\alpha }$ 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 $\alpha >1$ such that $\sigma$ is rational and further find that the existence of accumulation points explicitly depends on arithmetic properties of $\sigma$. Using these characterizations, we deduce existence and symmetry breaking results for ground state solutions of the reduced equation, extending known results.

36 pages, final version, to appear in Nonlinear Analysis