On the supercritical Schr\"{o}dinger equation on the exterior of a ball

We consider the mixed problem on the exterior of the unit ball in ${\mathbb{R}}^n$, $n\ge 2$, for a defocusing Schr"{o}dinger equation with a power nonlinearity $\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$, with zero boundary data. Assuming that the initial data are non radial, sufficiently small perturbations of \emph{large} radial initial data, we prove that for all powers $p>n+6$ the solution exists for all times, its Sobolev norms do not inflate, and the solution is unique in the energy class.