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

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.