  In this paper, we study the Gibbs measures associated to the focusing
nonlinear Schr\"odinger equation with harmonic potential on Euclidean spaces.
We establish a dichotomy for normalizability vs non-normalizability in the one
dimensional case, and under radial assumption in the higher dimensional cases.
In particular, we complete the programs of constructing Gibbs measures in the
presence of a harmonic potential initiated by Burq-Thomann-Tzvetkov (2005) in
dimension one and Deng (2013) in dimension two with radial assumption.
