Rotating waves in nonlinear media and critical degenerate Sobolev inequalities

Kübler, Joel; Weth, Tobias

doi:10.2140/apde.2025.18.307

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Rotating waves in nonlinear media and critical degenerate Sobolev inequalities

Joel Kübler and Tobias Weth

Vol. 18 (2025), No. 2, 307–360

DOI: 10.2140/apde.2025.18.307

Abstract

We investigate the presence of rotating wave solutions of the nonlinear wave equation $\partial_{t}^{2} v - Δ v + m v = | v |^{p - 2} v$ in $ℝ \times B$ , where $B \subset ℝ^{N}$ is the unit ball, complemented with Dirichlet boundary conditions on $ℝ \times \partial B$ . 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.

Keywords

nonlinear wave equation, degenerate Sobolev inequality, rotating solutions, symmetry-breaking, concentration-compactness

Mathematical Subject Classification

Primary: 35J20

Secondary: 35B33, 35J70

Milestones

Received: 7 April 2022

Revised: 6 October 2023

Accepted: 6 November 2023

Published: 5 February 2025

Authors

Joel Kübler
Institut für Mathematik Goethe-Universität Frankfurt am Main Germany

Tobias Weth
Institut für Mathematik Goethe-Universität Frankfurt am Main Germany

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Vol. 18, No. 2, 2025