Normalized grounded states for a coupled nonlinear schr\"{o}dinger system on $\mathbb{R}^3$

We investigate the existence of normalized ground states to the system of coupled Schr\"odinger equations: \begin{equation}\label{eq:0.1} \begin{cases} -\Delta u_1 + \lambda_1 u_1 = \mu_1 |u_1|^{p_1-2}u_1 + \beta r_1|u_1|^{r_1-2}u_1|u_2|^{r_2} & \text{ in } \mathbb{R}^{3}, -\Delta u_2 + \lambda_2 u_2 = \mu_2|u_2|^{p_2-2}u_2 + \beta r_2|u_1|^{r_1}|u_2|^{r_2-2}u_2 & \text{ in } \mathbb{R}^3, \end{cases} \end{equation} subject to the constraints $\mathcal{S}_{a_1} \times \mathcal{S}_{a_2} = \{(u_1 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_1^2 dx = a_1^2\} \times \{(u_2 \in H^1(\mathbb{R}^3))|\int_{\mathbb{R}^3} u_2^2 dx = a_2^2\}$, where $\mu_1, \mu_2 > 0$, $r_1, r_2 > 1$, and $\beta \geq 0$. Our focus is on the coupled mass super-critical case, specifically, $$\frac{10}{3} < p_1, p_2, r_1 + r_2 < 2^* = 6.$$ We demonstrate that there exists a $\tilde{\beta} \geq 0$ such that equation (\ref{eq:0.1}) admits positive, radially symmetric, normalized ground state solutions when $\beta > \tilde{\beta}$. Furthermore, this result can be generalized to systems with an arbitrary number of components, and the corresponding standing wave is orbitally unstable.