Nonlinear Schrödinger Equations on Periodic Metric Graphs

The paper is devoted to the nonlinear Schrödinger equation with periodic linear and nonlinear potentials on periodic metric graphs. Assuming that the spectrum of linear part does not contain zero, we prove the existence of finite energy ground state solution which decays exponentially fast at infinity. The proof is variational and makes use of the generalized Nehari manifold for the energy functional combined with periodic approximations. Actually, a finite energy ground state solution is obtained from periodic solutions in the infinite wave length limit.