Shift and Stability of Ground States of a Nonlinear Schrödinger Equation Outside a Small Insulated Domain

We study the behavior and stability of ground states of a Neumann problem in the exterior of a shrinking domain. We show that the ground states concentrate at a finite point which is a global maximum point of some function related to the domain. Then we use this result to prove the stability of the corresponding solitary waves of the time-dependent nonlinear Schrödinger equation.