In this work we consider the stability of localized structures in discrete nonlinear Schrödinger lattices with generalized nonlinearities, depending on the absolute value of the field. We illustrate how the continuation of solutions in one-, as well as higher dimensions proceeds from the anti-continuum limit and show how to generalize the results of Pelinovsky et al (2005 Physica D 212 1) for arbitrary nonlinearities. As a case example of particular experimental relevance, we showcase our main findings in the special setting of the lattice with the saturable (photorefractive) nonlinearity in one and two dimensions. Our analytical results are found to be in good agreement with direct numerical computations
