Rotating waves in nonlinear media and critical degenerate Sobolev inequalities

TL;DR

The paper analyzes rotating-wave solutions of a nonlinear wave equation on bounded radial domains by reducing to a semilinear elliptic problem with a rotating term $\alpha^2\partial_\theta^2$. It develops a family of degenerate, anisotropic Sobolev inequalities on $\mathbb{R}^N$ and in the ball, annulus, and Riemannian models, establishing existence and regularity of ground states and revealing symmetry-breaking transitions as the angular velocity $\alpha$ varies, with critical exponents $2_s^*$. A key thread is the link between the $\alpha=1$ degenerate case and half-space inequalities $\mathcal S_1(\mathbb{R}^N_+)$, which governs sharp thresholds and minimizer existence. The results extend to more general geometries, including annuli and hemispheres, and provide a unified variational approach to traveling/rotating waves via a broad class of associated critical degenerate Sobolev inequalities.

Abstract

We investigate the presence of rotating wave solutions of the nonlinear wave equation $\partial_t^2 v - Δv +m v = |v|^{p-2} v$ 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 $α$ 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 $α$, $m$ 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.

Theorems & Definitions (61)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• proof
• Theorem 2.2