Logarithmic Bose-Einstein condensates with harmonic potential

In this paper, by using a compactness method, we study the Cauchy problem of the logarithmic Schr"{o}dinger equation with harmonic potential. We then address the existence of ground states solutions as minimizers of the action on the Nehari manifold. Finally, we explicitly compute ground states (Gausson-type solution) and we show their orbital stability.