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arXiv · 1401.2242

Global well-posedness and scattering for nonlinear Schr\"odinger equations with combined nonlinearities in the radial case

Xing Cheng·Changxing Miao·Lifeng Zhao
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Abstract

We consider the Cauchy problem for the nonlinear Schr\"odinger 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 $H^1(\Bbb R^d)$ and then utilize the concentration-compactness method to show the global wellposedness and scattering versus blowup in $H^1(\Bbb R^d)$ below the threshold for radial data when $d\leq4$.

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Xing Cheng, Changxing Miao, Lifeng Zhao. 2014-01-10. Global well-posedness and scattering for nonlinear Schr\"odinger equations with combined nonlinearities in the radial case. https://doi.org/10.1016/j.jde.2016.04.031

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