Positive ground states for a system of Schr\"odinger equations with critically growing nonlinearities

We study the following problem $$\{\begin{array}{ll}-\mathrm{\Delta }u=\lambda u+u^{2^{\ast }-2}v& \text{in}\mathrm{\Omega },\\ -\mathrm{\Delta }v=\mu v^{2^{\ast }-1}+u^{2^{\ast }-1}& \text{in}\mathrm{\Omega },\\ u>0,v>0& \text{in}\mathrm{\Omega },\\ u=v=0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ is a bounded domain of ${\mathbb{R}}^N$, $N\ge 4$, $2^{\ast }=2N\mathrm{/}(N-2)$, $\lambda \in \mathbb{R}$ and $\mu \ge 0$ and we obtain existence and nonexistence results, depending on the value of the parameters $\lambda$ and $\mu$.