Normalized solutions to subcritical Choquard systems with double couplings

We consider the Choquard system with both linear and nonlinear couplings $-\Delta u + \mu_1 u =\lambda_1 ( I_\alpha * |u|^{r_1} ) |u|^{r_1-2} u + \beta p( I_\alpha * |v|^q)|u|^{p-2} u + \kappa v,$ $-\Delta v + \mu_2 v =\lambda_2 ( I_\alpha * |v|^{r_2} ) |v|^{r_2-2} v + \beta q( I_\alpha * |u|^p)|v|^{q-2} v + \kappa u ,$ $\int_{\mathbb{R}^N} u^2 = \rho_1^2\, , \int_{\mathbb{R}^N} v^2 = \rho_2^2,$ where $N \in \{3,4\}$, $\lambda_1, \lambda_2, \beta, \kappa, \rho_1,\rho_2 > 0$, $2_{\alpha,*} :=\frac{N+\alpha}{N} <p,q , r_1, r_2 <2_\alpha^*:=\frac{N+\alpha}{N-2}$ and $p+q\leq 2r_1 \leq 2r_2$ . We investigate a classification result as the parameters $p+q$, $2r_1$ and $2r_2$ vary across the ranges $(\frac{2N+2\alpha}{N},\frac{2N+2\alpha+4}{N})$, $\{\frac{2N+2\alpha+4}{N}\}$, and $(\frac{2N+2\alpha+4}{N},\frac{2N+2\alpha}{N-2})$. Employing variational methods, we demonstrate the existence of a normalized ground state for the system in the mass subcritical, critical, and supercritical cases.