  We consider the mixed problem on the exterior of the unit ball in
$\mathbb{R}^{n}$, $n\ge2$, for a defocusing Schr\"{o}dinger equation with a
power nonlinearity $|u|^{p-1}u$, 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.
