  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.
