Blow-up profile of the focusing gross-pitaevskii minimizer under self-gravitating effect

Abstract

We consider a Bose-Einstein condensate in a 2D dilute Bose gas, with an external potential and an interaction potential containing both 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.