Nonlinear Differential Equations and Applications (NoDEA) ( IF 1.2 ) Pub Date : 2021-09-03 , DOI: 10.1007/s00030-021-00722-7 Van Duong Dinh 1, 2
In this paper, we study a class of nonlinear Schrödinger equations (NLS) with potential
$$\begin{aligned} i\partial _t u +\Delta u - Vu = \pm |u|^\alpha u, \quad (t,x) \in \mathbb R\times \mathbb R^3, \end{aligned}$$where \(\frac{4}{3}<\alpha <4\) and V is a Kato-type potential including the genuine Yukawa potential as a special case. By using variational analysis and interaction Morawetz estimates, we establish a scattering criterion for the equation with non-radial initial data. As a consequence, we prove the energy scattering for the focusing problem with data below the ground state threshold. Our result extends the recent works of Hong (Commun Pure Appl Anal 15(5):1571–1601, 2016) and Hamano and Ikeda (J Evolut Equ 20:1131–1172, 2020). As a by product of the scattering criterion and the concentration-compactness lemma à la P. L. Lions, we study long time dynamics of global solutions to the focusing problem with data at the ground state threshold. Our result is robust and can be applicable to show the energy scattering for the focusing NLS with Coulomb potential.
中文翻译：
带势的非线性薛定谔方程的非径向散射理论
在本文中，我们研究了一类具有势能的非线性薛定谔方程（NLS）
$$\begin{aligned} i\partial _t u +\Delta u - Vu = \pm |u|^\alpha u, \quad (t,x) \in \mathbb R\times \mathbb R^3, \结束{对齐}$$其中\(\frac{4}{3}<\alpha <4\)和V是加藤型电位，包括真正的汤川电位作为特例。通过使用变分分析和相互作用 Morawetz 估计，我们为具有非径向初始数据的方程建立了散射准则。因此，我们用低于基态阈值的数据证明了聚焦问题的能量散射。我们的结果扩展了 Hong (Commun Pure Appl Anal 15(5):1571–1601, 2016) 和 Hamano 和 Ikeda (J Evolut Equ 20:1131–1172, 2020) 的近期工作。作为散射标准和浓度 - 紧凑性引理 à la PL Lions 的副产品，我们研究了聚焦问题的全局解决方案的长期动态，其数据处于基态阈值。我们的结果是稳健的，可用于显示具有库仑势的聚焦 NLS 的能量散射。
京公网安备 11010802027423号