  We explore the consequences of including the repulsive three-particle
interaction in the model of Bose-Einstein condensate dark matter model or fuzzy
dark matter. Such a model based on properly modified Gross-Pitaevskii equation
is intended to describe the distribution of dark matter particles in the highly
dense regions, which correspond to the galaxy core and/or to the overlap of
colliding galaxies. Specifically, we deal with the $\phi^6$-model in terms of
the macroscopic wave function of the condensate, where a locality of
interaction is guaranteed by a large correlation length assumed to hold. After
calculation of main thermodynamical characteristics, we find strong evidence of
the existence of two distinct phases of dark matter, within its core, separated
by the instability region lying between two differing special values of the
pressure acting in the model. Some implications stemming from the existence of
two phases and the related first-order phase transition are discussed.
