For a smooth, bounded domain
,
,
we
consider the nonlocal equation
with zero Dirichlet datum and a small parameter
.
We construct a family of solutions that concentrate as
at an interior
point of the domain in the form of a scaling of the ground state in entire space. Unlike the
classical case
,
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
in the expanding
domain
with zero exterior datum.
Keywords
nonlocal quantum mechanics, Green functions, concentration
phenomena