## Scattering for the mass super-critical perturbations of the mass critical nonlinear Schrödinger equations.(English)Zbl 1435.35344

Summary: We consider the Cauchy problem for the nonlinear Schrödinger (NLS) equation with double nonlinearities with opposite sign, with one term mass-critical and the other term mass-supercritical and energy-subcritical, which includes the well-known two-dimensional cubic-quintic NLS equation arising in the study of the boson gas with 2- and 3-body interactions. We prove global well-posedness and scattering in $$H^1(\mathbb{R}^d)$$ below the threshold for nonradial data when $$1\le d\le 4$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q41 Time-dependent Schrödinger equations and Dirac equations 35L70 Second-order nonlinear hyperbolic equations 35P25 Scattering theory for PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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