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Blow-up profile of the focusing Gross-Pitaevskii minimizer under self-gravitating effect. (English) Zbl 1448.35424

Summary: 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.

MSC:

35Q40 PDEs in connection with quantum mechanics
46N50 Applications of functional analysis in quantum physics
49J20 Existence theories for optimal control problems involving partial differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35B44 Blow-up in context of PDEs
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