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This paper deals with the study of "\textit{sharp localized}" solutions of anonlinear 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 themodulus of the solution, and with a non zero external forcing term $\f.$ Ourfundamental assumption is that such an exponent $m$ verifies $m\in (0,1).$ Theself-similar structure of the solution is justified from the assumption thatthe 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 isassumed to be with compact support in $\R^N.$ We show the existence ofsolutions $\vu(t,x)=t^{\vp/2}\U(t^{-1/2}x),$ with a profile $\U,$ which alsohave compact support in $\R^N,$ reason why we call as "\textit{sharplocalized}" solutions to this type of solutions. The proof of the localizationof the support of the profile $\U$ uses some suitable energy method applied tothe stationary problem satisfied by $\U$ after some unknown transformation.

Self-similar solutions with compactly supported profile of some nonlinear Schr{\"o}dinger equations