  This paper deals with the study of "\textit{sharp localized}" solutions of a
nonlinear type Schr{\"o}dinger equation in the whole space $\R^N,$ $N\ge1,$
with a zero order term, in modulus, like a power $m$ less than one of the
modulus of the solution, and with a non zero external forcing term $\f.$ Our
fundamental assumption is that such an exponent $m$ verifies $m\in (0,1).$ The
self-similar structure of the solution is justified from the assumption that
the external forcing term satisfies that $\f(t,x)=t^{-(\vp-2)/2}\F(t^{-1/2}x)$
for some complex exponent $\vp$ and for some profile function $\F$ which is
assumed to be with compact support in $\R^N.$ We show the existence of
solutions $\vu(t,x)=t^{\vp/2}\U(t^{-1/2}x),$ with a profile $\U,$ which also
have compact support in $\R^N,$ reason why we call as "\textit{sharp
localized}" solutions to this type of solutions. The proof of the localization
of the support of the profile $\U$ uses some suitable energy method applied to
the stationary problem satisfied by $\U$ after some unknown transformation.
