Ground States of a 𝐾-Component Critical System with Linear and Nonlinear Couplings: The Attractive Case

Consider the system \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\geq 2} , {\Omega\subset\mathbb{R}^{N}} ( {N\geq 3} ) is a bounded domain, {2^{*}=\frac{2N}{N-2}} , {\mu_{i}\in\mathbb{R}} and {\nu_{i}>0} are constants, and {\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 {\beta,\lambda} are also established.