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Transactions of the American Mathematical Society

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Ground states of two-component attractive Bose-Einstein condensates II: Semi-trivial limit behavior


Authors: Yujin Guo, Shuai Li, Juncheng Wei and Xiaoyu Zeng
Journal: Trans. Amer. Math. Soc. 371 (2019), 6903-6948
MSC (2010): Primary 35J50, 35J47; Secondary 46N50
DOI: https://doi.org/10.1090/tran/7540
Published electronically: February 14, 2019
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Abstract: As a continuation of our prior article, we study new pattern formations of ground states $ (u_1,u_2)$ for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in $ \mathbb{R}^2$, where the intraspecies interaction $ (-a,-b)$ and the interspecies interaction $ -\beta $ are both attractive, i.e., $ a$, $ b$, and $ \beta $ are all positive. If $ 0<b<a^*:=\Vert w\Vert^2_2$ and $ 0<\beta <a^*$ are fixed, where $ w$ is the unique positive solution of $ \Delta w-w+w^3=0$ in $ \mathbb{R}^2$, the semi-trivial behavior of $ (u_1,u_2)$ as $ a\nearrow a^*$ is proved in the sense that $ u_1$ concentrates at a unique point and while $ u_2\equiv 0$ in $ \mathbb{R}^2$. However, if $ 0<b<a^*$ and $ a^*\le \beta <\beta ^*=a^*+\sqrt {(a^*-a)(a^*-b)}$, the refined spike profile and the uniqueness of $ (u_1,u_2)$ as $ a\nearrow a^*$ are analyzed, where $ (u_1,u_2)$ must be unique, $ u_1$ concentrates at a unique point, and meanwhile $ u_2$ can either blow up or vanish, depending on how $ \beta $ approaches $ a^*$.


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Additional Information

Yujin Guo
Affiliation: School of Mathematics and Statistics, and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, People’s Republic of China
Email: yjguo@wipm.ac.cn

Shuai Li
Affiliation: University of Chinese Academy of Sciences, Beijing 100190, People’s Republic of China –and– Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, People’s Republic of China
Email: lishuai_wipm@outlook.com

Juncheng Wei
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Email: jcwei@math.ubc.ca

Xiaoyu Zeng
Affiliation: Department of Mathematics, Wuhan University of Technology, Wuhan 430070, People’s Republic of China
Email: xyzeng@whut.edu.cn

DOI: https://doi.org/10.1090/tran/7540
Keywords: Gross-Pitaevskii equations, ground states, minimizers, mass concentration, semi-trivial solutions
Received by editor(s): August 15, 2017
Received by editor(s) in revised form: November 21, 2017, and December 23, 2017
Published electronically: February 14, 2019
Additional Notes: The first and second authors were partially supported by NSFC under Grant No. 11671394 and by MOST under Grant No. 2017YFA0304500.
The third author was partially supported by NSERC of Canada.
The fourth author was partially supported by NSFC grant 11501555.
Article copyright: © Copyright 2019 American Mathematical Society