We discuss the existence of breather solutions for a discrete nonlinear Schrödinger equation in an infinite $N$-dimensional lattice, involving site-dependent anharmonic parameters. We give a simple proof of the existence of (non-trivial) breather solutions based on a variational approach, assuming that the sequence of anharmonic parameters is in an appropriate sequence space (decays with an appropriate rate). We also give a proof of the non-existence of (non-trivial) breather solutions, and discuss a possible physical interpretation of the restrictions, in both the existence and non-existence cases.
