  We propose existence and multiplicity results for the system of Schr\"odinger
equations with sign-changing nonlinearities in bounded domains or in the whole
space $\mathbb{R}^N$. In the bounded domain we utilize the classical approach
via the Nehari manifold, which is (under our assumptions) a differentiable
manifold of class $\mathcal{C}^1$ and the Fountain theorem by Bartsch. In the
space $\mathbb{R}^N$ we additionally need to assume the
$\mathbb{Z}^N$-periodicity of potentials and our proofs are based on the
concentration-compactness lemma by Lions and the Lusternik-Schnirelmann values.
