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Focusing and defocusing cases of the purely elliptic generalized Davey-Stewartson system. (English) Zbl 1185.35254
Summary: We define the focusing and the defocusing cases for the purely elliptic generalized Davey-Stewartson system. These cases are mutually exclusive and exhaustive and therefore close the gap that was left in the previous studies. In the defocusing case, all solutions exist globally. In the focusing case, any initial data can be scaled to one with negative energy. The solution with the scaled initial data then blows up in finite time. We also show the existence of standing waves and the global existence and scattering of solutions with subminimal mass. Our results equally apply to the elliptic almost-cubic non-linear Schrödinger equation as described in [Commun. Pure Appl. Anal. 8, No. 6, 1803–1823 (2009; Zbl 1180.35480)].
MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35B44Blow-up (PDE)
35P25Scattering theory (PDE)