Existence and asymptotic behavior of normalized solutions for coupled critical Choquard equations

In this paper, we study the coupled critical Choquard equations with prescribed mass \begin{equation} \begin{aligned} \left\{ \begin{array}{lll} -\Delta u+\lambda_1u=(I_\mu\ast |u|^{2^*_\mu})|u|^{2^*_\mu-2}u+\nu p(I_\mu\ast |v|^q)|u|^{p-2}u\ & \text{in}\quad \mathbb{R}^N,\\ -\Delta v+\lambda_2v=(I_\mu\ast |v|^{2^*_\mu})|v|^{2^*_\mu-2}v+\nu q(I_\mu\ast |u|^p)|v|^{q-2}v\ & \text{in}\quad \mathbb{R}^N,\\ \int_{\mathbb{R}^N}u^2=a^2,\quad\int_{\mathbb{R}^N}v^2=b^2, \end{array}\right.\end{aligned} \end{equation} where $N\in\{3,4\}$, $0<\mu0$, we study the existence, non-existence and qualitative behavior of normalized solutions by distinguishing three cases: $L^2$-subcritical case: $p+q<4+\frac{4-2\mu}{N}$; $L^2$-critical case: $p+q=4+\frac{4-2\mu}{N}$; $L^2$-supercritical case: $p+q>4+\frac{4-2\mu}{N}$. In particular, in $L^2$-subcritical and $L^2$-critical cases, we show the system has a radial normalized ground state for $0<\nu<\nu_0$ with $\nu_0$ explicitly given. In $L^2$-supercritical case, we prove the existence of two thresholds $\nu_2\geq\nu_1\geq0$ such that a radial normalized solution exists if $\nu>\nu_2$, and no normalized ground state exists for $\nu<\nu_1$. Moreover, we give the concrete ranges of $p$ and $q$ for $\nu_2=\nu_1=0$ and $\nu_2\geq\nu_1>0$.