CONCENTRATION PHENOMENA FOR THE NONLOCAL SCHRODINGER EQUATION WITH DIRICHLET DATUM

For a smooth, bounded domain Omega, s is an element of (0, 1), P is an element of (1, (n+2s)/(n-2s)) we consider the nonlocal equation epsilon(2s)(-Delta)(s)u + u = up in Omega with zero Dirichlet datum and a small parameter epsilon > 0. We construct a family of solutions that concentrate as epsilon -> 0 at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case s = 1, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function of epsilon(2s)(-Delta)(s) + 1 in the expanding domain epsilon(-1)Omega with zero exterior datum.