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Existence and local uniqueness of normalized solutions for two-component Bose–Einstein condensates

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Abstract

In this paper, we consider the following two-component Bose–Einstein condensates (BEC)

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1 +V_1(x) u_1= a_1 u_1^3 +\mu u_1+\beta u_1u_2^2 &{}~\text {in}\;\mathbb R^2,\\ \\ -\Delta u_2+V_2(x) u_2= a_2 u_2^3 +\mu u_2+\beta u_1^2u_2 &{}~\text {in}\;\mathbb R^2, \end{array}\right. } \end{aligned}$$

with the constraint \(\displaystyle \int \limits _{\mathbb R^2}(u_1^2+u_2^2)dx=1\). The existence and local uniqueness of k-peak solutions are given by finite-dimensional reduction and local Pohozaev identities, which gives the description of excited state of BEC phenomenon stated and generalizes the results in [15] about the ground states.

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Acknowledgements

The authors would like to thank Shuangjie Peng and Peng Luo for interesting discussion on local uniqueness. Qing Guo has been supported by NSFC Grants (No.11771469).

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Authors and Affiliations

  1. College of Science, Minzu University of China, Beijing, 100081, China

    Qing Guo

  2. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Huafei Xie

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  1. Qing Guo
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Guo, Q., Xie, H. Existence and local uniqueness of normalized solutions for two-component Bose–Einstein condensates. Z. Angew. Math. Phys. 72, 189 (2021). https://doi.org/10.1007/s00033-021-01619-2

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