  We consider interacting Bose particles in an external potential. It is shown
that a Bose-Einstein condensate is possible at finite temperatures that
describes a supersolid in three dimensions (3D) for a wide range of potentials
in the absence of an external potential. However, for 2D, a self-organized
supersolid exists for finite temperatures provided the interaction between
bosons is nonlocal and of infinitely long-range. It is interesting that in the
absence of the latter type of potential and in the presence of a lattice
potential, there is no Bose-Einstein condensate and so in such a case, a 2D
supersolid is not possible at finite temperatures. We also propose the correct
Bloch form of the condensate wave function valid for finite temperatures, which
may be used as the correct trial wave function.
