  The main goal of this paper is to study the nature of the support of the
solution of suitable nonlinear Schr\"{o}dinger equations mainly the compactness
of the support and its spatial localization. This question is very related with
pure essence of the derivation of the Schr\"{o}dinger equation since it is
well-known that if the linear Schr\"{o}dinger equation is perturbated with
"regular" potentials then the corresponding solution never vanishes on a
positive measured subset of the domain, which corresponds with the
impossibility of localize the particle. Here we shall prove that if the
perturbation involves suitable singular nonlinear terms then the support of the
solution becomes a compact set and so any estimate on its spatial localization
implies a very rich information on places which can not be. Our results are
obtained by the application of some energy methods which connect the
compactness of the support with the local vanishing of a suitable "energy
function" which satisfies a nonlinear differential inequality with an exponent
less than one. The results improve and extend a previous short presentation by
the authors published in 2006.
