Existence, nonexistence, symmetry and uniqueness of ground state for critical Schr\"odinger system involving Hardy term

We study the following elliptic system with critical exponent: \begin{displaymath} \begin{cases}-\Delta u_j-\frac{\lambda_j}{|x|^2}u_j=u_j^{2^*-1}+\sum\limits_{k\neq j}\beta_{jk}\alpha_{jk}u_j^{\alpha_{jk}-1}u_k^{\alpha_{kj}},\;\;x\in\R^N, u_j\in D^{1,2}(\R^N),\quad u_j>0 \;\; \hbox{in} \quad \R^N\setminus \{0\},\quad j=1,...,r.\end{cases}\end{displaymath} Here $N\geq 3, r\geq2, 2^*=\frac{2N}{N-2}, \lambda_j\in (0, \frac{(N-2)^2}{4})$ for all $ j=1,...,r $; $\beta_{jk}=\beta_{kj}$; \; $\alpha_{jk}>1, \alpha_{kj}>1,$ satisfying $\alpha_{jk}+\alpha_{kj}=2^* $ for all $k\neq j$. Note that the nonlinearities $u_j^{2^*-1}$ and the coupling terms all are critical in arbitrary dimension $N\geq3 $. The signs of the coupling constants $\bb_{ij}$'s are decisive for the existence of the ground state solutions. We show that the critical system with $r\geq 3$ has a positive least energy solution for all $\beta_{jk}>0$. However, there is no ground state solutions if all $\beta_{jk}$ are negative. We also prove that the positive solutions of the system are radially symmetric. Furthermore, we obtain the uniqueness theorem for the case $r\geq 3$ with $N=4$ and the existence theorem when $r=2$ with general coupling exponents.