Blow-up profile of the focusing Gross-Pitaevskii minimizer under self-gravitating effect

TL;DR

This paper investigates the blow-up behavior of minimizers in a 2D Bose-Einstein condensate model with combined short-range attraction and long-range self-gravitating effects, identifying the universal blow-up profile as a Gagliardo-Nirenberg optimizer.

Contribution

It establishes the existence of minimizers and characterizes their blow-up profile at the critical interaction strength, linking it to a Gagliardo-Nirenberg inequality optimizer.

Findings

Existence of minimizers for the model.

Identification of the universal blow-up profile.

Connection to Gagliardo-Nirenberg inequality optimizer.

Abstract

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.