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Existence and symmetry of positive ground states for a doubly critical Schrödinger system


Authors: Zhijie Chen and Wenming Zou
Journal: Trans. Amer. Math. Soc. 367 (2015), 3599-3646
MSC (2010): Primary 35J50, 35J47; Secondary 35B33, 35B09
DOI: https://doi.org/10.1090/S0002-9947-2014-06237-5
Published electronically: September 19, 2014
MathSciNet review: 3314818
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Abstract: We study the following doubly critical Schrödinger system:

$\displaystyle \begin {cases}-\Delta u -\frac {\lambda _1}{\vert x\vert^2}u=u^{2... ...b{R}^N),\quad u,\, v>0\,\,\hbox {in $\mathbb{R}^N\setminus \{0\}$},\end{cases} $

where $ N\ge 3$, $ \lambda _1, \lambda _2\in (0, \frac {(N-2)^2}{4})$, $ 2^\ast =\frac {2N}{N-2}$ and $ \alpha >1, \beta >1$ satisfying $ \alpha +\beta =2^\ast $. This problem is related to coupled nonlinear Schrödinger equations with critical exponent for Bose-Einstein condensate. For different ranges of $ N$, $ \alpha $, $ \beta $ and $ \nu >0$, we obtain positive ground state solutions via some quite different variational methods, which are all radially symmetric. It turns out that the least energy level depends heavily on the relations among $ \alpha , \,\beta $ and $ 2$. Besides, for sufficiently small $ \nu >0$, positive solutions are also obtained via a variational perturbation approach. Note that the Palais-Smale condition cannot hold for any positive energy level, which makes the study via variational methods rather complicated.

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Additional Information

Zhijie Chen
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: chenzhijie1987@sina.com

Wenming Zou
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: wzou@math.tsinghua.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2014-06237-5
Received by editor(s): December 13, 2012
Received by editor(s) in revised form: July 2, 2013
Published electronically: September 19, 2014
Additional Notes: This work was supported by NSFC (11025106, 11371212, 11271386) and the Both-Side Tsinghua Fund.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.