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This paper investigates dipolar Bose‐Einstein condensate modeled by the Gross‐Pitaevskii equation with harmonic confinement and dipolar interaction potential. With the help of a new estimate of the kinetic energy as well as a mechanical analogy, the known results of collapse[14] have been extended to a new ample condition for the existence of blowup solutions.
The main purpose of this paper is to study the existence of extremal functions for the singular Trudinger–Moser inequalities in the critical case in . More precisely, let and denote
then we will prove in this article that there exists such that can be achieved for all , .
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