The focusing NLS on exterior domains in three dimensions

Abstract

We consider the Dirichlet problem of the focusing energy subcritical NLS outside a smooth compact strictly convex obstacle in dimension three. The critical space of our problem is $\dot{H}^s$ with $0<s<1$. In this paper, we proved that if the initial data $u_{0}$ satisfy $\Vert u_{0}\Vert _{2}^{1-s}\Vert \nabla u_{0}\Vert _{2}^{s}<\Vert \nabla Q\Vert _{2}^{s}\Vert Q\Vert _{2}^{1-s}$ and $ M(u_{0})^{1-s}E(u_{0})^{s}<M(Q)^{1-s}E(Q)^{s},$ then there exists a unique global solution which scatters in both time directions, where $Q$ denotes the ground state solution in the whole space case.