Rotating waves in nonlinear media and critical degenerate Sobolev inequalities

TL;DR

This paper studies rotating wave solutions of nonlinear wave equations in bounded domains, linking their existence to new critical degenerate Sobolev inequalities, and characterizes conditions for nonradial ground states in various geometries.

Contribution

It establishes a connection between rotating wave solutions and critical degenerate Sobolev inequalities, providing existence criteria and analyzing nonradial solutions in diverse geometric settings.

Findings

Existence of ground state solutions depends on parameters α, m, p.

Derived new critical degenerate Sobolev inequalities in half space.

Identified conditions for nonradial, truly rotating wave solutions.

Abstract

We investigate the presence of rotating wave solutions of the nonlinear wave equation $\partial_t^2 v - \Delta v +m v = |v|^{p-2} v$ in $\mathbb{R} \times \mathbf{B}$, where $\mathbf{B} \subset \mathbb{R}^N$ is the unit ball, complemented with Dirichlet boundary conditions on $\mathbb{R} \times \partial\mathbf{B}$. Depending on the prescribed angular velocity $\alpha$ of the rotation, this leads to a Dirichlet problem for a semilinear elliptic or degenerate elliptic equation. We show that this problem is governed by an associated critical degenerate Sobolev inequality in the half space. After proving this inequality and the existence of associated extremal functions, we then deduce necessary and sufficient conditions for the existence of ground state solutions. Moreover, we analyze under which conditions on $\alpha$, $m$ and $p$ these ground states are nonradial and therefore give rise to truly rotating waves. Our approach carries over to the corresponding Dirichlet problems in an annulus and in more general Riemannian models with boundary, including the hemisphere. We briefly discuss these problems and show that they are related to a larger family of associated critical degenerate Sobolev inequalities.