Consider the following system of N coupled Schrödinger equations \( \left\{ \begin{aligned}&-\Delta u_j+V_j(x)u_j=\sum \nolimits _{i=1}^N\beta _{ij}(x)u_i^2u_j \quad \text {in}\ {\mathbb {R}}^n, \\&\,u_j\in H^1({\mathbb {R}}^n),\quad j=1, 2, \ldots , N, \end{aligned} \right. \) which arises in the theory of Bose–Einstein condensates and nonlinear optics. Here, \(n=1, 2, 3\) , \(N\ge 2\) , \(\beta _{ij}=\beta _{ji}\) for \(i, j=1, 2, \ldots , N\) , and the potentials \(V_j\) and \(\beta _{ij}\) are either periodic or asymptotically periodic functions. When the coupling potentials \(\beta _{ij}\,(i\ne j)\) are relatively small compared with the self-interaction potentials \(\beta _{ii}\) and \(\beta _{jj}\) , we prove that the system has a fully nontrivial positive ground state using variational method. Since the system has many semi-trivial solutions, careful avoidance of their energy levels is essential in constructing a fully nontrivial solution.
