In this paper, we consider the Cauchy problem for nonlinear Schrödinger 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 for the equation. These results extend a recent work of Miao-Zhang-Zheng (2018, arXiv:1809.06685) to a general class of inverse-power potentials and higher dimensions.

中文翻译：

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具有排斥性逆功率势的非线性Schrödinger方程

在本文中，我们考虑具有排斥反功率势的非线性薛定ö方程的柯西问题

$$ i \ partial _ {t} u + \ Delta u-c | x | ^ {-\ sigma} u = \ pm | u | ^ {\ alpha} u，\ quad c> 0。$$

我们研究方程的能量空间中的局部和全局适定性，有限时间爆炸和散射。这些结果将Miao-Zhang-Zheng（2018，arXiv：1809.06685）的最新工作扩展到了反功率势和更高维度的一般类别。