Consider the system:
\begin{equation*}
\left\{\aligned&-\Delta u_i+\mu_i u_i=\nu_iu_i^{2^*-1}+\beta\sum_{j=1,j\not=i}^ku_j^{\frac{2^*}{2}}u_i^{\frac{2^*}{2}-1}+\lambda \sum_{j=1,j\not=i}^ku_j\quad\text{in }\Omega,\\
&u_i>0\quad\text{in }\Omega,\\
&u_i=0\quad\text{on }\partial\Omega,\quad i=1,2,\cdots,k,\endaligned\right.
\end{equation*}
where
,
\Omega\subset\bbr^N(N\geq3) is a bounded
... [Show full abstract] domain, , \mu_i\in\bbr and are constants, and are parameters. By showing a unique result of the limit system, we prove existence and nonexistence results of ground states to this system by variational methods, which generalize the results in \cite{CZ131,PSW17}. Concentration behaviors of ground states for are also established.