  We solve a minimization problem related to the cubic Lowest Landau level
equation, which is used in the study of Bose-Einstein condensation. We provide
an optimal condition for the Gaussian to be the unique global minimizer. This
extends previous results from P. G{\'e}rard, P. Germain and L. Thomann. We then
provide another condition so that the second special Hermite function is a
global minimizer.
