In this paper, we study the following critical system with fractional Laplacian:
$\begin{equation*}\begin{cases}(-Δ)^{s}u = μ_{1}|u|^{2^{*}-2}u+\dfrac{αγ}{2^{*}}|u|^{α-2}u|v|^{β} \ \ \ \text{in} \ \ \mathbb{R}^{n},\\(-Δ)^{s}v = μ_{2}|v|^{2^{*}-2}v+\dfrac{βγ}{2^{*}}|u|^{α}|v|^{β-2}v\ \ \ \ \text{in} \ \ \mathbb{R}^{n},\\u,v∈ D_{s}(\mathbb{R}^{n}).\end{cases}\end{equation*}$
By using the Nehari manifold, under proper conditions, we establish the existence and nonexistence of positive least energy solution of the system.
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