In this paper, we study the following perturbation problem with Sobolev critical exponent: $$\left\{ {\begin{array}{*{20}{c}}{ - \Delta u = (1 + \varepsilon K(x)){u^{2* - 1}} + \frac{\alpha }{{{2^*}}}{u^{\alpha - 1}}{v^\beta } + \varepsilon h(x){u^p},}&{x \in {\mathbb{R}^N},} \\{ - \Delta v = (1 + \varepsilon Q(x)){v^{2* - 1}} + \frac{\beta }{{{2^*}}}{u^\alpha }{v^{\beta - 1}} + \varepsilon l(x){v^q},}&{x \in {\mathbb{R}^N},} \\{u > 0,v > 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\...
