I will focus on the existence and multiplicity of positive non-radial solutions for some systems of Schrödinger equations in a weak fully attractive or repulsive regime in presence or in absence of an external radial trapping. The results have been obtained in collaboration with Haixia Chen (Central China Normal University, Wuhan), Maria Medina (Universidad Autonoma de Madrid ) and Giusi Vaira (Università di Bari).