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In this paper, we consider the following fractional Laplacian system with onecritical exponent and one subcritical exponent \begin{equation*} \begin{cases}  (-\Delta)^{s}u+\mu u=|u|^{p-1}u+\lambda v & x\in \ \mathbb{R}^{N},  (-\Delta)^{s}v+\nu v = |v|^{2^{\ast}-2}v+\lambda u& x\in \ \mathbb{R}^{N},\\\end{cases} \end{equation*} where $(-\Delta)^{s}$ is the fractional Laplacian,$0<s<1,\ N>2s, \ \lambda <\sqrt{\mu\nu },\ 1<p<2^{\ast}-1~ and~\2^{\ast}=\frac{2N}{N-2s}$~ is the Sobolev critical exponent. By using theNehari\ manifold, we show that there exists a $\mu_{0}\in(0,1)$, such that when$0<\mu\leq\mu_{0}$, the system has a positive ground state solution. When$\mu>\mu_{0}$, there exists a$\lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\sqrt{\mu\nu})$ such that if$\lambda>\lambda_{\mu,\nu}$, the system has a positive ground state solution,if $\lambda<\lambda_{\mu,\nu}$, the system has no ground state solution.

Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent