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

We study the following doubly critical Schr\"{o}dinger system $$-\Delta u -\frac{\la_1}{|x|^2}u=u^{2^\ast-1}+ \nu \al u^{\al-1}v^\bb, \quad x\in \RN, -\Delta v -\frac{\la_2}{|x|^2}v=v^{2^\ast-1} + \nu \bb u^{\al}v^{\bb-1}, \quad x\in \RN, u, v\in D^{1, 2}(\RN),\quad u, v>0 in $\RN\setminus{0}$},$$ where $N\ge 3$, $\la_1, \la_2\in (0, \frac{(N-2)^2}{4})$, $2^\ast=\frac{2N}{N-2}$ and $\al>1, \bb>1$ satisfying $\al+\bb=2^\ast$. This problem is related to coupled nonlinear Schr\"{o}dinger equations with critical exponent for Bose-Einstein condensate. For different ranges of $N$, $\al$, $\bb$ and $\nu>0$, we obtain positive ground state solutions via some quite different methods, which are all radially symmetric. It turns out that the least energy level depends heavily on the relations among $\al, \bb$ and 2. Besides, for sufficiently small $\nu>0$, positive solutions are also obtained via a variational perturbation approach. Note that the Palais-Smale condition can not hold for any positive energy level, which makes the study via variational methods rather complicated.