On nonlinear Schrödinger equations with repulsive inverse-power potentials

Abstract

In this paper, we consider the Cauchy problem for the nonlinear Schrödinger equations with repulsive inverse-power potentialsWe 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 Schrödinger equation with coulomb potential, arXiv:1809.06685] to a general class of inverse-power potentials and higher dimensions.

Global dynamics for a class of inhomogeneous nonlinear Schrödinger equations with potential

“…The proof is based on the interaction Morawetz inequality. A similar result for the repulsive inverse-power potentials was established in [13]. Let us start with the following classical Morawetz inequality.…”

On nonlinear Schrödinger equations with attractive inverse-power potentials

“…This paper is a continuation of [13] where nonlinear Schrödinger equations with repulsive (i.e. the minus sign in front of |x| −σ u) inverse-power potentials in the energy space were considered.…”

“…the minus sign in front of |x| −σ u) inverse-power potentials in the energy space were considered. The local well-posedness (LWP) for (1.1) in the energy space H 1 was studied in [13]. More precisely, the author showed that (1.1) is locally well-posed in H 1 for both energy-subcritical and energy-critical cases.…”

“…In fact, by radial Sobolev embeddings (see e.g. [13]), it suffices to show that for ε > 0 small enough, there exists δ(ε) > 0 such that…”

Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions

“…We have from [11,Proposition 4.7] (by taking V = 0 and W = |x| −b ) that the following interaction Morawetz inequality holds true for the defocusing problem (1.1) in dimensions N ≥ 3…”