A remark on the existence of breather solutions for the Discrete Nonlinear Schr\"odinger Equation in infinite lattices: The case of site dependent anharmonic parameter

We discuss the existence of breather solutions for a Discrete Nonlinear Schr\"odinger equation in an infinite $N$-dimensional lattice, involving site dependent anharmonic parameter. We give a simple proof on the existence of (nontrivial) breather solutions based on variational approach, assuming that the sequence of anharmonic parmeters is in an appropriate sequence space (decays with an appropriate rate). We also give a proof on the non-existence of (non-trivial) breather solutions, and discuss a possible physical interpretation of the restrictions, both in the existence and nonexistence case.