Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents

In this paper we are interested in the following critical coupled Hartree system \begin{document}$ \left\{\begin{array}{l} (-\Delta)^{s} \tilde{u}+\lambda_{1}\tilde{u} = \alpha_{1}\int_{\Omega}\frac{|\tilde{u}(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}dz|\tilde{u}|^{2_{\mu}^{\ast}-2}\tilde{u}\\ \quad +\beta\int_{\Omega}\frac{|\tilde{v}(z)|^{2_{\mu}^{\ast}}} {|x-z|^{\mu}}dz|\tilde{u}|^{2_{\mu}^{\ast}-2}\tilde{u}, \ \ &&\mbox{in} \ \Omega,\\ (-\Delta)^{s} \tilde{v}+\lambda_{2}\tilde{v} = \alpha_{2}\int_{\Omega}\frac{|\tilde{v}(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}dz|\tilde{v}|^{2_{\mu}^{\ast}-2}\tilde{v}\\ \quad +\beta\int_{\Omega}\frac{|\tilde{u}(z)|^{2_{\mu}^{\ast}}} {|x-z|^{\mu}}dz|\tilde{v}|^{2_{\mu}^{\ast}-2}\tilde{v}, \ \ &&\mbox{in} \ \Omega,\\ \tilde{u} = \tilde{v} = 0, \ \ && \mbox{on} \ \partial \Omega, \end{array} \right. $\end{document} where \begin{document}$ 0 , \begin{document}$ \alpha_{1}, \alpha_{2}>0 $\end{document} , \begin{document}$ \beta\neq0 $\end{document} , \begin{document}$ 4s , \begin{document}$ 2_{\mu}^{\ast} = (2N-\mu)/(N-2s) $\end{document} , \begin{document}$ \Omega\subset\mathbb{R}^N(N\geq3) $\end{document} is a smooth bounded domain, \begin{document}$ -\lambda_{1}(\Omega) with \begin{document}$ \lambda_{1}(\Omega) $\end{document} the first eigenvalue of \begin{document}$ (-\Delta)^{s} $\end{document} under the Dirichlet boundary condition. Assume that the nonlinearity and the coupling terms are both of the upper critical growth due to the Hardy–Littlewood–Sobolev inequality, by applying the Dirichlet-to-Neumann map, we are able to obtain the existence of the ground state solution of the critical coupled Hartree system.