Breather solutions of the discrete nonlinear Schrodinger equations with unbounded potentials

In this paper I investigate the existence of nontrivial breather solutions of the discrete nonlinear Schrodinger equation with the unbounded potential at infinity. First I derive a discrete version of compact embedding theorem. Then combining the Nehari manifold approach and the compact embedding theorem, I show the existence of breather solutions without Palais-Smale condition. The results on the exponential decay of breather solutions are also provided in this paper.