In this work we consider a system of nonlinear Schrödinger equations whose nonlinearities satisfy a power-type-growth. First, we prove that the Cauchy problem is local and global well-posedness in \(L^2\) and \(H^1\) . Next, we establish the existence of ground state solutions. Then we use these solutions to study the dichotomy of global existence versus blow-up in finite time. Similar results were presented in the reference Noguera and Pastor (Commun Contemp Math 23:2050023, 2021. https://doi.org/10.1142/S0219199720500236) for the special case when the growth of the nonlinearities was quadratic. Here we will extend them to systems with nonlinearities of order p (cubic, quartic and so on). Finally, we recover some known results for two particular systems, one with quadratic and the other with cubic growth nonlinearities.
