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Ground state solution for weakly coupled time-harmonic Maxwell’s equations with critical exponent

Part of: Equations of mathematical physics and other areas of application Elliptic equations and systems Partial differential equations Optics, electromagnetic theory - General

Published online by Cambridge University Press:  18 July 2025

Zhen Song
Zhen Song*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, China (sz21@mails.tsinghua.edu.cn)
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Abstract

In this paper, we deal with the following nonlinear time-harmonic Maxwell’s equations

\begin{equation*}\nonumber\begin{cases} \nabla \times(\nabla \times u_1)=\mu_1|u_1|^4u_1+\beta|u_1||u_2|^3u_1 & \text {in } \mathbb{R}^3, \\ \nabla \times(\nabla \times u_2)=\mu_2|u_2|^4u_2+\beta|u_1|^3|u_2|u_2 & \text {in } \mathbb{R}^3, \end{cases}\end{equation*}
where $\nabla\times$ denotes the usual curl operator in $\mathbb{R}^3$, $\mu_1,\mu_2 \gt 0$, and $\beta\in\mathbb{R}\backslash\{0\}$. We show that this critical system admits a non-trivial ground state solution when the parameter β is positive and small. For general $\beta\in\mathbb{R}\backslash\{0\}$, we prove that this system admits a non-trivial cylindrically symmetric solution with the least positive energy. We also study the existence of the curl-free solution and the synchronized solution due to the special structure of this system. These seem to be the first results on the critically coupled system containing the curl-curl operator.

Keywords

critically coupled systemcylindrically symmetric solutionground state solutionnonlinear time-harmonic Maxwell’s equations

MSC classification

Primary: 35Q60: PDEs in connection with optics and electromagnetic theory
Secondary: 35J20: Variational methods for second-order elliptic equations 35B33: Critical exponents 78A25: Electromagnetic theory, general

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 31
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

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