Summary: In this paper, we give a different method from [Y. Ao and W. Zou, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 181, 222–248 (2019; Zbl 1418.35193)] and [Z. Chen and W. Zou, J. Funct. Anal. 262, No. 7, 3091–3107 (2012; Zbl 1234.35241)] to consider the following nonlinear Schrödinger system with one critical exponent and one subcritical exponent: \[\begin{cases} - \varDelta_p u + \mu | u |^{p - 2} u = | u |^{q - 2} u + \alpha \lambda | u |^{\alpha - 2} u | v |^\beta & \text{in } \mathbb{R}^N, \\ - \varDelta_p v + \nu | v |^{p - 2} v = | v |^{p^\ast - 2} v + \beta \lambda | u |^\alpha | v |^{\beta - 2} v & \text{in } \mathbb{R}^N, \end{cases}\] where \(N \geq \max \{ 3 , p \}\), \(\mu , \nu , \lambda > 0\) , \(\alpha \geq 1\) , \(\beta \geq 1\) , \(2 \leq p < q < p^\ast\) and \(\alpha + \beta = p\) , \(p^\ast = N p / ( N - p )\). By using variational methods, we prove that there exists \(\mu_0 \in ( 0 , 1 )\), such that when \(0 < \mu \leq \mu_0\), the above system has a positive ground state solution; when \(\mu > \mu_0\), there exists \[\lambda_{\mu , \nu} \in \left[ \Big( \frac{ \mu - \mu_0}{ \alpha}\Big)^{\frac{ \alpha}{ p}} \Big( \frac{ \nu}{ \beta}\Big)^{\frac{ \beta}{ p}} ,\Big(\frac{ \mu}{ \alpha} \Big)^{\frac{ \alpha}{ p}} \Big( \frac{ \nu}{ \beta}\Big)^{\frac{ \beta}{ p}}\right)\] such that if \(\lambda > \lambda_{\mu , \nu} \), the above system has a positive ground state solution, if \(\lambda < \lambda_{\mu , \nu} \), the above system has no ground state solution.