  In this paper we consider the NLS equation with power nonlinearity and a
point interaction (a "$\delta$-potential" in the physical literature) in
dimension two and three. We will show that for low power nonlinearities there
is failure of scattering to the free dynamics or to standing waves. In the
recent paper Murphy and Nakanishi consider the NLS equation with potentials and
measures, singular enough to include the $\delta$-potential in dimension one
and they show analogous properties. In this respect our contribution is an
extension to higher dimension and it needs a different treatment of the linear
part of the interaction, due the qualitatively different and stronger character
of the singularity involved.
