We consider the following nonlinear Schrödinger system with mixed couplings in \mathbb {R}^3: \begin{equation*} -\Delta u_i + \lambda _i u_i=\mu _i u_i^3+\sum \limits _{j=1,j\neq i}^N\beta _{ij}u_j^2u_i,\ \ \ i=1,\cdots ,N, \end{equation*} where \lambda _i, \mu _i>0, \beta _{ij}=\beta _{ji} (i,j=1,\cdots ,N, i\neq j). The system appears in modeling of Bose-Einstein condensates theory. While most existing works in the literature are concerned with purely attractive or purely repulsive couplings (i.e., all \beta _{ij} have the same signs), we ...
