### Abstract

Consider the system {-Δui+μiui=νiui2*-1+β∑j=1,j≠ikuj2*2ui2*2-1+λ∑j=1,j≠ikujinΩ,ui>0inΩ,ui=0on∂Ω,i=1,2,…,k,\left\{\begin{aligned} \displaystyle-\Delta u_{i}+\mu_{i}u_{i}&\displaystyle=% \nu_{i}u_{i}^{2^{*}-1}+\beta\mathop{\sum_{j=1,j\neq i}^{k}}u_{j}^{\frac{2^{*}}% {2}}u_{i}^{\frac{2^{*}}{2}-1}+\lambda\mathop{\sum_{j=1,j\neq i}^{k}}u_{j}&&% \displaystyle\phantom{}\text{in}\ \Omega,\\ \displaystyle u_{i}&\displaystyle>0&&\displaystyle\phantom{}\text{in}\ \Omega,% \\ \displaystyle u_{i}&\displaystyle=0&&\displaystyle\phantom{}\text{on}\ % \partial\Omega,\quad i=1,2,\ldots,k,\end{aligned}\right. where k≥2{k\geq 2}, Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} (N≥3{N\geq 3}) is a bounded domain, 2*=2NN-2{2^{*}=\frac{2N}{N-2}}, μi∈ℝ{\mu_{i}\in\mathbb{R}} and νi>0{\nu_{i}>0} are constants, and β,λ>0{\beta,\lambda>0} 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 [7, 18]. Concentration behaviors of ground states for β,λ{\beta,\lambda} are also established.