Existence and orbital stability of standing waves to nonlinear Schrödinger system with partial confinement

Abstract: We are concerned with the existence of solutions to the following nonlinear Schr\"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. Our solutions are achieved as global minimizers of the underlying energy functional subject to the constraint. Our purpose is to establish the compactness of any minimizing sequence up to translations. As a by-product, we obtain the orbital stability of the set of global minimizers.