We study the following elliptic system with critical exponent:
$$\left\{ {\begin{array}{*{20}{c}}
{ - \Delta u = {\lambda _1}u + {u_1}|u{|^{2*-2}}u + \beta |u{{|^{\frac{{2*}}{2} - 2}{{u|v|}}^{\frac{{2*}}{2}}}},\;\;x \in \Omega } \\
{ - \Delta v = {\lambda _2}v + {u_2}|v{|^{2*-2}}v + \beta |v{{|^{\frac{{2*}}{2} - 2}{{v|u|}}^{\frac{{2*}}{2}}}},\;\;x \in \Omega } \\ \;\;\;\;\;\;\;
{u = v = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \partial \Omega ,}
\end{array}} \right.\;\;$$
where...
