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Published online by Cambridge University Press: 18 July 2025
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*}
$\nabla\times$ 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.