  We consider a Bose-Einstein condensate in a 2D dilute Bose gas, with an
external potential and an interaction potential containing both of the
short-range attractive self-interaction and the long-range self-gravitating
effect. We prove the existence of minimizers and analyze their behavior when
the strength of the attractive interaction converges to a critical value. The
universal blow-up profile is the unique optimizer of a Gagliardo-Nirenberg
interpolation inequality.
