Article

Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations

Pascal Bégout, and Jesus Ildefonso Diaz

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.

Replaces

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.

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.

See also

Published in

Electronic Journal of Differential Equations, vol. 90, 2014, pp. 1–15