Ground states of a system of nonlinear Schr\"odinger equations with periodic potentials

We are concerned with a system of coupled Schr"odinger equations $$-\mathrm{\Delta }{u}_i+V_i(x)u_i={\mathrm{\partial }}_{u_i}F(x,u)\text{ on }{\mathbb{R}}^N,i=1,2,\mathrm{.}\mathrm{.}\mathrm{.},K,$$ where $F$ and $V_i$ are periodic in $x$ and $0\mathrm{\notin }\sigma (-\mathrm{\Delta }+V_i)$ for $i=1,2,\mathrm{.}\mathrm{.}\mathrm{.},K$, where $\sigma (-\mathrm{\Delta }+V_i)$ stands for the spectrum of the Schr"odinger operator $-\mathrm{\Delta }+V_i$. We impose general assumptions on the nonlinearity $F$ with the subcritical growth and we find a ground state solution being a minimizer of the energy functional associated with the system on a Nehari-Pankov manifold. Our approach is based on a new linking-type result involving the Nehari-Pankov manifold.