This paper deals with the system with linearly coupled of nonlocal problem with critical exponent, $ \begin{equation*} \begin{cases} (-\Delta)^\alpha u+\lambda_1u = |u|^{2_\alpha^*-2}u+\beta v , \quad x\in \Omega , \\ (-\Delta)^\alpha v+\lambda_2v = |v|^{2_\alpha^*-2}v+\beta u , \,\quad x\in \Omega , \\ u = v = 0,\ \ \qquad \qquad \qquad \quad \quad \qquad \,x\in \partial\Omega. \end{cases} \end{equation*} $ Here $ \Omega $ is a smooth bounded domain in $ {\mathbb{R}}^N(N>4\alpha) $, $ 00 $. Via a perturbation argument, by doing some delicate estimates for the nonlocal term, we overcome some d...
