We prove sharp existence and nonexistence results for minimal energy solutions of the nonlinear Schrödinger system 1 $$\left.\begin{array}{ll}\quad -\Delta u + u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q \quad {\rm in} \, \mathbb{R}^{n},\\ -\Delta v + \omega^2 v = |v|^{2q-2}v + b|u|^q|v|^{q-2}v \quad {\rm in} \, \mathbb{R}^{n}\end{array}\right.$$ - Δ u + u = | u | 2 q - 2 u + b | u | q - 2 u | v | q in R n , - Δ v + ω 2 v = | v | 2 q - 2 v + b | u | q | v | q - 2 v in R n in the cooperative and subcritical case $${b > 0, 1 < q < \frac{n}{(n-2)_+}}$$ b > 0 , 1 < q < n ( n - 2 ) + . The proofs are accomplished by minimizing the Euler functional of (1) over the two associated Nehari manifolds. In the special case $${1 < q < 2}$$ 1 < q < 2 we find that a positive solution of (1) with minimal energy among all nontrivial solutions exists if and only if b > 0.