  We study the existence of nontrivial solutions for a class of asymptotically
periodic semilinear Schr\"odinger equations in $\mathbb{R}^N$. By combining
variational methods and the concentration-compactness principle we obtain a
nontrivial solution for asymptotically periodic problem and a ground state
solution for the periodic problem. In the proofs we apply the Mountain Pass
Theorem and its local version.
