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Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement

  • * Corresponding author: Gongbao Li

    * Corresponding author: Gongbao Li 

The project is supported by National Natural Science Foundation of China(Grant No.11771166), Hubei Key Laboratory of Mathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University #IRT_17R46. H.F Jia is also supported by National Postdoctoral Science Foundation of China(Grant No.2019M662833). X. Luo is also supported by National Natural Science Foundation of China(Grant No.11901147)

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  • In this article, we deal with the existence, qualitative and symmetry properties of normalized solutions to the following nonlinear Schrödinger system

    $ \begin{equation*} \begin{cases} -\Delta u+(x_{1}^{2}+x_{2}^{2})u = \lambda_{1}u+\mu_{1}u^{3}+\beta uv^{2}, &\quad x\in \mathbb{R}^3,\\ -\Delta v+(x_{1}^{2}+x_{2}^{2})v = \lambda_{2}v+\mu_{2}v^{3}+\beta u^{2}v, &\quad x\in \mathbb{R}^3, \end{cases} \end{equation*} $

    where $ \mu_{i}>0 $ ($ i $ = 1, 2), $ \beta>0 $, and the frequencies $ \lambda_{1} $, $ \lambda_{2} $ are unknown and appear as Lagrange multipliers. In addition, we study the stability of the corresponding standing waves for the related time-dependent Schrödinger systems. We mainly extend the results in J. Bellazzini et al. (Commun. Math. Phys. 2017), which dealt with mass-supercritical nonlinear Schrödinger equation with partial confinement, to cubic nonlinear Schrödinger systems with partial confinement.

    Mathematics Subject Classification: Primary: 35J20, 35J25, 35J60.


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