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On nonlinear Schr\"odinger equations with repulsive inverse-power potentials
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abstract
In this paper, we consider the Cauchy problem for the nonlinear Schr\"odinger equations with repulsive inverse-power potentials \[ i \partial_t u + \Delta u - c |x|^{-\sigma} u = \pm |u|^\alpha u, \quad c>0. \] We study the local and global well-posedness, finite time blow-up and scattering in the energy space $H^1$ for the equation. These results extend a recent work of Miao-Zhang-Zheng [Nonlinear Schr\"odinger equation with coulomb potential, arXiv:1809.06685] to a general class of inverse-power potentials and higher dimensions.
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Energy scattering for a class of inhomogeneous nonlinear Schr\"odinger equation in two dimensions
For the 2D inhomogeneous NLS with 0<b<1 and α>2-b, radial H^1 solutions scatter in both focusing (below ground state) and defocusing cases.
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