  In this paper we consider a non-linear Schroedinger equation with a cubic
nonlinearity and a multi-dimensional double well potential. In the
semiclassical limit the problem of the existence of stationary solutions simply
reduces to the analysis of a finite dimensional Hamiltonian system which
exhibits different behavior depending on the dimension. In particular, in
dimension 1 the symmetric stationary solution shows a standard pitchfork
bifurcation effect, while in dimension 2 and 3 new asymmetrical solutions
associated to saddle points occur. These last solutions are localized on a
single well and this fact is related to the phase transition effect observed in
Bose-Einstein condensates in periodical lattices.
