Mass concentration behavior of Bose–Einstein condensates with attractive interactions in bounded domains

ABSTRACT

We study the Bose–Einstein condensates with attractive interaction in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$, described by Gross–Pitaevskii energy functional. We prove that there exists a constant $a^{\ast }>0$, such that minimizers exist if and only if $0<a<a^{\ast }$. As $a\nearrow a^{\ast }$, the limit behavior of minimizers is also analyzed, where the mass must concentrate at the global minimum points of the trapping potential $V\left(x\right)$. In particular, if $V\left(x\right)$ has the flattest global minimum points both at the interior and the boundary of Ω, we prove that the mass must concentrate at an inner minimum point of Ω, rather than the neighborhood of the boundary minimum points.

KEYWORDS:

• Bose–Einstein condensates
• bounded domains
• minimizers
• mass concentration

AMS SUBJECT CLASSIFICATIONS:

• 35Q40
• 46N50
• 82D50

Acknowledgements

The authors are very grateful to Professor Yujin Guo for his fruitful discussions in the present paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was supported in part by National Natural Science Foundation of China (NSFC) [11671394]. Xincai Zhu is also supported by Nanhu Scholars Program for Young Scholars of XYNU.