In this talk, we are concerned with the investigation of a class of linearly coupled Schr ̈odinger systems with steep potential well, which arises from Bose- Einstein condensates. The existence of positive ground states is investigated by exploiting the relation between the Nehari manifold and fibering maps. Some interesting phenomena are that we do not need the weight functions in the nonlinear terms to be integrable or bounded and we can relax the upper control condition of the coupling function. This is a joint work with Prof. Tsung-fang Wu and Prof. Ying-Chieh Lin.