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 Schrödinger system in $\mathbb{R}^3$: \begin{equation*} \left\{ \begin{aligned} -Δu_1 + (x_1^2+x_2^2)u_1&= λ_1 u_1 + μ_1 |u_1|^{p_1 -2}u_1 + βr_1|u_1|^{r_1-2}u_1|u_2|^{r_2}, \\ -Δu_2 + (x_1^2+x_2^2)u_2&= λ_2 u_2 + μ_2 |u_2|^{p_2 -2}u_2 +βr_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 $μ_1, μ_2, β>0, 2 1, r_1 + r_2 < \frac{10}{3}$. In the system, the parameters $λ_1, λ_2 \in \R$ 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.