Reference

Pascal Bégout, and Jesus Ildefonso Diaz, Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations, Electronic Journal of Differential Equations, vol. 90, 2014, pp. 1–15.

Abstract

This paper deals with the study of "\textit{sharp localized}" solutions of a nonlinear type Schr\"odinger 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.

Pascal Bégout, and Jesus Ildefonso Diaz, Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations, TSE Working Paper, n. 13-400, April 2013.