Abstract
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 p > 1 + 4 N?
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