Critical system involving fractional Laplacian

Abstract: In this paper, we study the following critical system with fractional Laplacian: \begin{equation*} \begin{cases} (-\Delta)^{s}u= \mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha\gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} \ \ \ \text{in} \ \ \mathbb{R}^{n}, (-\Delta)^{s}v= \mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta\gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v\ \ \ \ \text{in} \ \ \mathbb{R}^{n}, u,v\in 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.