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We are concerned with the existence of solutions to the following nonlinearSchr\"odinger system in $\mathbb{R}^3$: \begin{equation*} \left\{\begin{aligned} -\Delta u_1 + (x_1^2+x_2^2)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}, \\ -\Delta u_2 + (x_1^2+x_2^2)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, \end{aligned} \right.\end{equation*} under the constraint \begin{align*} \int_{\mathbb{R}^3}|u_1|^2\, dx = a_1>0,\quad \int_{\mathbb{R}^3}|u_2|^2 \, dx = a_2>0, \end{align*}where $\mu_1, \mu_2, \beta >0, 2 <p_1, p_2 < \frac{10}{3}$, $r_1, r_2>1, r_1 +r_2 < \frac{10}{3}$. In the system, the parameters $\lambda_1, \lambda_2 \in\R$ are unknown and appear as the associated Lagrange multipliers. Oursolutions are achieved as global minimizers of the underlying energy functionalsubject to the constraint. Our purpose is to establish the compactness of anyminimizing sequence up to translations. As a by-product, we obtain the orbitalstability of the set of global minimizers.

Existence and orbital stability of standing waves to nonlinear Schr\"odinger system with partial confinement