In this paper, we will study the following fractional system with critical growth \(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s} u + V(x)u = \displaystyle \frac{1}{2^{*}_{s}}Q_{u}(u,v)\quad \text {in } \mathbb {R}^{N},\\ \\ (-\Delta )^{s} v + W(x)v= \displaystyle \frac{1}{2^{*}_{s}}Q_{v}(u,v)\quad \text {in } \mathbb {R}^{N}, \end{array}\right. } \end{aligned}\) where \(s\in (0,1)\) , \(N \ge 4s\) , \(2^{*}_{s}=\displaystyle \frac{2N}{N-2s}\) and V and W are potential functions. We combine a recent global compactness result by Correia and Oliveira [1] with Krasnoselskii’s genus theory to demonstrate that the system has at least N distinct pairs of non-trivial solutions in the case of small perturbations of the potentials.
