Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data

We consider the defocusing energy-critical nonlinear Schrödinger (NLS) in the exterior of the unit ball in three dimensions.For the initial value problem with Dirichlet boundary condition, we prove global well-posedness and scattering with large radial initial data in the Sobolev space Ḣ1 0 .We also point out that the same strategy can be used to treat the energy-supercritical NLS in the exterior of balls with Dirichlet boundary condition and radial Ḣ1 0 initial data.