We study the following system of non-linear Schrodinger equations where Ω Rn (n ≥ 2) is a bounded smooth domain, and ai bi di si ε R and pi = pi1, . . ., pik are constants (for notation, see the first paragraph below). Under appropriate conditions, we derive a priori estimates and existence of positive solutions for arbitrary k ≥ 2, extending the earlier results of  for k = 2 to all k≥2. © Taylor & Francis Group, LLC.