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Transactions of the American Mathematical Society

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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: \[ \begin {cases}-\Delta u -\frac {\lambda _1}{|x|^2}u=u^{2^\ast -1}+ \nu \alpha u^{\alpha -1}v^\beta , \quad x\in \mathbb {R}^N,\\ -\Delta v -\frac {\lambda _2}{|x|^2}v=v^{2^\ast -1} + \nu \beta u^{\alpha }v^{\beta -1}, \quad x\in \mathbb {R}^N,\\ u, v\in D^{1, 2}(\mathbb {R}^N),\quad u, v>0 \hbox {in $\mathbb {R}^N\setminus \{0\}$},\end {cases} \] 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
MR Author ID: 366305
Email: wzou@math.tsinghua.edu.cn

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.