Standing waves of a weakly coupled Schrödinger system with distinct potential functions

The standing wave solutions of a weakly coupled nonlinear Schrödinger system with distinct trapping potential functions in RNRN (1≤N≤31≤N≤3) are considered. This type of system arises from models in Bose–Einstein condensates theory and nonlinear optics. The existence of a positive ground state solution is shown when the coupling constant is larger than a sharp threshold value, which is explicitly defined in terms of potential functions and system parameters. It is also shown that such solutions concentrate near the minimum points of potential functions, and multiple positive concentration solutions exist when the topological structure of the set of minimum points satisfies certain condition. Variational approach is used for the existence and concentration of positive solutions.