  We discuss the existence of breathers and lower bounds on their power, in
nonlinear Schr\"odinger lattices with nonlinear hopping. Our methods extend
from a simple variational approach to fixed point arguments, deriving lower
bounds for the power which can serve as a threshold for the existence of
breather solutions. Qualitatively, the theoretical results justify
non-existence of breathers below the prescribed lower bounds of the power which
depend on the dimension, the parameters of the lattice as well as of the
frequency of breathers. In the case of supercritical power nonlinearities we
investigate the interplay of these estimates with the optimal constant of the
discrete interpolation inequality. Improvements of the general estimates,
taking into account the localization of the true breather solutions are
derived. Numerical studies in the one dimensional lattice corroborate the
theoretical bounds and illustrate that in certain parameter regimes of physical
significance, the estimates can serve as accurate predictors of the breather
power and its dependence on the various system parameters.
