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 ${\varphi }^6$-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.