Electron. J. Diff. Equ., Vol. 2014 (2014), No. 90, pp. 1-15.

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

Pascal Begout, Jesus Ildefonso Diaz

``Sharp localized'' solutions (i.e. with compact support for each given time t) of a singular nonlinear type Schr\"odinger equation in the whole space $\mathbb{R}^N$ are constructed here under the assumption that they have a self-similar structure. It requires the assumption that the external forcing term satisfies that $\mathbf{f}(t,x)=t^{-(\mathbf{p}-2)/2}\mathbf{F}(t^{-1/2}x)$ for some complex exponent $\mathbf{p}$ and for some profile function $\mathbf{F}$ which is assumed to be with compact support in $\mathbb{R}^N$. We show the existence of solutions of the form , with a profile $\mathbf{U}$, which also has compact support in $\mathbb{R}^N$. The proof of the localization of the support of the profile $\mathbf{U}$ uses some suitable energy method applied to the stationary problem satisfied by $\mathbf{U}$ after some unknown transformation.

Submitted December 9, 2013. Published April 2, 2014.
Math Subject Classifications: 35B99, 35A01, 35A02, 35B65, 35J60.
Key Words: Nonlinear self-similar Schrodinger equation; compact support; energy method.

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Pascal Bégout
Institut de Mathématiques de Toulouse & TSE
Université Toulouse I Capitole, Manufacture des Tabacs
21, Allée de Brienne, 31015 Toulouse Cedex 6, France
email: Pascal.Begout@math.cnrs.fr
Jesús Ildefonso Díaz
Departamento de Matemática Aplicada
Instituto de Matemática Interdisciplinar
Universidad Complutense de Madrid
Plaza de las Ciencias, 3, 28040 Madrid, Spain
email: diaz.racefyn@insde.es

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