Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means of variational argument. We establish the profile decomposition in $H^1({\mathbb{R}}^d)$ and then utilize the concentration-compactness method to show the global wellposedness and scattering versus blowup in $H^1({\mathbb{R}}^d)$ below the threshold for radial data when $d\le 4$.

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