SELF-SIMILAR SOLUTIONS WITH COMPACTLY SUPPORTED PROFILE OF SOME NONLINEAR SCHRODINGER EQUATIONS

"Sharp localized" solutions (i.e. with compact support for each given time t) of a singular nonlinear type Schrodinger equation in the whole space 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 f (t, x) = t F(p-2)/2(t (1/2x)) for some complex exponent p and for some profile function F which is assumed to be with compact support in R-N. We show the existence of solutions of the form u(t, x) = t(P/2)U(t(-1/2x)), with a profile U, which also has compact support in R-N. 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.