  In the present work we revisit the issue of the self-trapping dynamical
transition at a nonlinear impurity embedded in an otherwise linear lattice. For
our Schr\"odinger chain example, we present rigorous arguments that establish
necessary conditions and corresponding parametric bounds for the transition
between linear decay and nonlinear persistence of a defect mode. The proofs
combine a contraction mapping approach applied in the fully dynamical problem
in the case of a 3D-lattice, together with variational arguments for the
derivation of parametric bounds for the creation of stationary states
associated with the expected fate of the self-trapping dynamical transition.
The results are relevant for both power law nonlinearities and saturable ones.
The analytical results are corroborated by numerical computations.
