In this paper, we study the following Dirichlet problem with Sobolev critical exponent
$$\begin{aligned} \left\{ \begin{array}{ll}\displaystyle -\Delta u=|u|^{2^*-2}u+\displaystyle \frac{\alpha }{2^*}|u|^{\alpha -2}|v|^{\beta }u,&{}\quad x\in \Omega , \\ -\Delta v=|v|^{2^*-2}v+\displaystyle \frac{\beta }{2^*}|u|^{\alpha }|v|^{\beta -2}v,&{}\quad x\in \Omega , \end{array} \right. \end{aligned}$$
where
$$\alpha , \beta >1,$$
$$\alpha +\beta =2^*:=\frac{2N}{N-2}(N\ge 3)$$
and
$$\Omega ={\mathbb {R}}^N$$
or
$$\Omega $$
is...
