This paper concerns an elliptic system with critical exponents: -Δuj-λj|x|2uj=uj2∗-1+∑k≠jβjkαjkujαjk-1ukαkj,x∈RN,uj∈D1,2(RN),uj>0inRN\{0},j=1,…,r,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_j-\frac{\lambda _j}{|x|^2}u_j=u_j^{2^*-1}+\sum \limits _{k\ne j}\beta _{jk}\alpha _{jk}u_j^{\alpha _{jk}-1}u_k^{\alpha _{kj}},\;\;x\in {{\mathbb {R}}}^N,\\ u_j\in D^{1,2}({{\mathbb {R}}}^N),\quad u_j>0 \;\; \hbox {in} \quad {{\mathbb {R}}}^N\setminus \{0\},\quad j=1, \ldots ,r,\end{array}\right. } \end{aligned}$$\end{document}where N≥3,r≥2,2∗=2NN-2,λj∈(0,(N-2)24)\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 3, r\ge 2, 2^*=\frac{2N}{N-2}, \lambda _j\in (0, \frac{(N-2)^2}{4})$$\end{document} for all j=1,…,r\documentclass[12pt]{minimal}
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\begin{document}$$ j=1, \ldots ,r $$\end{document}; βjk=βkj\documentclass[12pt]{minimal}
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\begin{document}$$\beta _{jk}=\beta _{kj}$$\end{document}; αjk>1,αkj>1,\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{jk}>1, \alpha _{kj}>1,$$\end{document} and αjk+αkj=2∗\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _{jk}+\alpha _{kj}=2^* $$\end{document} for all k≠j\documentclass[12pt]{minimal}
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\begin{document}$$k\ne j$$\end{document}. Note that the nonlinearities uj2∗-1\documentclass[12pt]{minimal}
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\begin{document}$$u_j^{2^*-1}$$\end{document} and the coupling terms are all critical in arbitrary dimension N≥3\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 3 $$\end{document}. The signs of the coupling constants βij\documentclass[12pt]{minimal}
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\begin{document}$$\beta _{ij}$$\end{document} are decisive for the existence of the ground-state solutions. We show that the critical system with r≥3\documentclass[12pt]{minimal}
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\begin{document}$$r\ge 3$$\end{document} has a positive ground-state solution for all βjk>0\documentclass[12pt]{minimal}
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\begin{document}$$\beta _{jk}>0$$\end{document} with some constraint on λj\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _j$$\end{document}. However, there is no ground-state solution when all βjk\documentclass[12pt]{minimal}
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\begin{document}$$\beta _{jk}$$\end{document} are negative. It is also proved that the positive solution of the system is radially symmetric. Furthermore, we obtain an uniqueness theorem for the case r≥3\documentclass[12pt]{minimal}
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\begin{document}$$r\ge 3$$\end{document} with N=4\documentclass[12pt]{minimal}
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\begin{document}$$N=4$$\end{document} and an existence theorem for the case r=2\documentclass[12pt]{minimal}
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\begin{document}$$r=2$$\end{document} with general coupling exponents.
