Sharp threshold of global existence for non-linear Gross–Pitaevskii equation in ?N

摘要

This paper is concerned with the non-linear Gross–Pitaevskii equation which describes the attractive Bose–Einstein condensate under a magnetic trap. By an intricate variational argument we derive out a sharp threshold of blowing up and global existence by applying the potential well argument and the concavity method. Furthermore, we answer the question: How small are the initial data, the global solutions of the Cauchy problem of the equation exist for