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 (R d ) and then utilize the concentration-compactness method to show the global wellposedness and scattering versus blowup in H 1 (R d ) below the threshold for radial data when d ≤ 4.

In this article, we will show the global wellposedness and scattering of the cubic defocusing nonlinear Schrödinger equation on waveguide R 2 ×T in H 1 . We first establish the linear profile decomposition in H 1 (R 2 × T) motivated by the linear profile decomposition of the mass-critical Schrödinger equation in L 2 (R 2 ). Then by using the solution of the infinite dimensional vector-valued resonant nonlinear Schrödinger system to approximate the nonlinear profile, we can prove scattering in H 1 by using the concentrationcompactness/rigidity method.

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