Abstract
We consider two-dimensional Bose–Einstein condensates with inhomogeneous attractive interactions \(0<m(x)\le 1\), which can be described by the Gross–Pitaevskii functional. We prove that minimizers exist if and only if the interaction strength \(a\) satisfies \(a < a^*= \Vert Q\Vert _2^2\), where \(Q\) is the unique positive radial solution of \(\Delta u-u+u^3=0\) in \({\mathbb {R}}^2\). The concentration behavior and symmetry breaking of minimizers as \(a\) approaches \(a^*\) are also analyzed, where all the mass concentrates at a global minimum point \(x_0\) of the trapping potential \(V(x)\), provided that \(x_0\) is also a global maximum point of \(m(x)\).
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Acknowledgments
The authors are grateful to Prof. Robert Seiringer for his fruitful discussions on the subject. The authors would also like to thank the referee for the thorough review and helpful comments which helped us revise and improve this article.
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Communicated by A. Malchiodi.
Y. B. Deng is partially supported by NSFC Grant No. 11371160, as well as a program for Changjiang Scholars and Innovative Research Team in University No. IRT13066. Y. J. Guo is partially supported by NSFC Grant No. 11322104, as well as the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
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Deng, Y., Guo, Y. & Lu, L. On the collapse and concentration of Bose–Einstein condensates with inhomogeneous attractive interactions. Calc. Var. 54, 99–118 (2015). https://doi.org/10.1007/s00526-014-0779-9
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DOI: https://doi.org/10.1007/s00526-014-0779-9