Blow-up profile of Bose-Einstein condensate with singular potentials

Abstract

The paper is concerned with the Bose-Einstein condensate described by the attractive Gross-Pitaevskii equation in R 2 , where the external potential is unbounded from below. We show that when the interaction strength increases to a critical value, the Gross-Pitaevskii minimizer collapses to one singular point and we analyze the details of the collapse exactly up to the leading order.

Blow-up Profile of the Focusing Gross-Pitaevskii Minimizer Under Self-Gravitating Effect

“…For example, if the external potential V is not singular enough, then the self-gravitating interaction is the main cause of the instability and the details of the blow-up phenomenon are more or less irrelevant to V . This situation is very different from the case without gravity studied in [11,12,22,19]. The precise form of our results will be provided in the next section.…”

“…This is an interesting effect of the self-gravitating interaction. In the case without gravity studied in [11,12,22,19], the blow-up behavior depends crucially on the local behavior of V around its minimizers/singular points, which can be seen from (7). Heuristically, if the condensate shrinks with a length scale ε → 0, then the self-gravitating interaction is of order ε −1 , while the external potential is of order at most ε −p (since V is not singular than |x| −p ).…”

Ground state of the mass-critical inhomogeneous nonlinear Schrödinger functional

“…The most difficult part is the third inequality in (21). Inspired by Guo-Seiringer [3] (see also [13]), we will prove that a substantial part of the mass of u a concentrates close to the minima x j of K. However, the perturbation method in [3,13] does not work in our case and we have to develop new ideas in the proof below.…”

Asymptotic behavior of normalized ground states for the fractional Schrödinger equation with combined L2‐critical and L2‐subcritical nonlinearities

“…After this work, problems of this type have been extensively studied; see previous studies. [22][23][24][25][26][27][28][29][30][31][32][33][34][35] Motivated by the above aforementioned papers, we study the asymptotic behavior of minimizers for (7) as c ↗ c * . We can obtain the following results.…”