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Nonlinear Schrödinger equation with Coulomb potential

Changxing Miao, Junyong Zhang, Jiqiang Zheng

Abstract

In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential $i\partial_tu+Δu+\frac{K}{|x|}u=λ|u|^{p-1}u$ with $1<p\leq5$ on $\mathbb{R}^3$. We mainly consider the influence of the long range potential $K|x|^{-1}$ on the existence theory and scattering theory for nonlinear Schrödinger equation. In particular, we prove the global existence when the Coulomb potential is attractive, i.e. $K>0$ and scattering theory when the Coulomb potential is repulsive i.e. $K\leq0$. The argument is based on the interaction Morawetz-type inequalities and the equivalence of Sobolev norms.

Nonlinear Schrödinger equation with Coulomb potential

Abstract

In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential with on . We mainly consider the influence of the long range potential on the existence theory and scattering theory for nonlinear Schrödinger equation. In particular, we prove the global existence when the Coulomb potential is attractive, i.e. and scattering theory when the Coulomb potential is repulsive i.e. . The argument is based on the interaction Morawetz-type inequalities and the equivalence of Sobolev norms.

Paper Structure

This paper contains 14 sections, 15 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Strichartz estimate
  5. Fractional product rule
  6. Sobolev space equivalence
  7. Global well-posedness
  8. Local well-posedness for energy-subcritical: $s_c<1$
  9. Global well-posedness for energy-subcritical: $s_c<1$
  10. Global well-posedness for energy-critical: $s_c=1$ and $K<0$
  11. Morawetz estimate and scattering theory
  12. Morawetz estimate
  13. Scattering theory
  14. Blow up

Key Result

Theorem \oldthetheorem

Let $K\in\mathbb{R}$ and $u_0\in H^1(\mathbb{R}^3)$. Suppose that $0<p-1\leq4$ in the defocusing case $\lambda=1$. While for the focusing case $\lambda=-1$, we assume that $0<p-1<\frac{4}{3}$$($mass-subcritical$)$ or Then, there exists a unique global solution $u(t,x)$ to equ1.1 such that for any $I\subset\mathbb{R}$ compact and $(q, r)\in\Lambda_0$ admissible defined below.

Theorems & Definitions (25)

  • Theorem \oldthetheorem: Global well-posedness
  • Remark \oldthetheorem
  • Theorem \oldthetheorem: Scattering theory
  • Theorem \oldthetheorem: Blow-up result
  • Proposition \oldthetheorem: Hölder's inequality in Lorentz space
  • Theorem \oldthetheorem: Local-in-time Strichartz estimate
  • proof
  • Lemma \oldthetheorem: Strichartz estimate for $e^{it\Delta}$, KTPlan
  • Theorem \oldthetheorem: Global-in-time Strichartz estimate,Miz
  • Lemma \oldthetheorem: Fractional product rule
  • ...and 15 more