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The uniqueness of the ground state and the dynamics of nonlinear Schrödinger equation with inverse square potential

Kai Yang, Chongchun Zeng, Xiaoyi Zhang

Abstract

In this paper, we first provide an alternative proof of the uniqueness of the ground state solution for NLS with inverse square potential and power nonlinearity $|u|^pu$ for all $0<p<\frac 4{d-2}$ in dimensions $d\ge 3$. While the uniqueness result was previously obtained by Mukherjee-Nam-Nguyen using a functional analytic approach, our method successfully adapts the classical ``shooting method'' to the case with the singular potential, accompanied by a more detailed analysis on the ground state equation. Based upon this result and a comprehensive spectral analysis, we construct the stable/unstable manifolds of the ground state standing wave solutions and classify solutions on the mass-energy level surface of the ground state in dimensions $d=3, 4, 5$.

The uniqueness of the ground state and the dynamics of nonlinear Schrödinger equation with inverse square potential

Abstract

In this paper, we first provide an alternative proof of the uniqueness of the ground state solution for NLS with inverse square potential and power nonlinearity for all in dimensions . While the uniqueness result was previously obtained by Mukherjee-Nam-Nguyen using a functional analytic approach, our method successfully adapts the classical ``shooting method'' to the case with the singular potential, accompanied by a more detailed analysis on the ground state equation. Based upon this result and a comprehensive spectral analysis, we construct the stable/unstable manifolds of the ground state standing wave solutions and classify solutions on the mass-energy level surface of the ground state in dimensions .
Paper Structure (24 sections, 32 theorems, 438 equations)

This paper contains 24 sections, 32 theorems, 438 equations.

Table of Contents

  1. Introduction
  2. Existence of Ground States and the parametrization of local $H^1$ solutions
  3. The existence of the ground state solution
  4. Asymptotics of local $H^1_{rad}$ solutions of equation \ref{['2']} as $r\to +\infty$
  5. Asymptotics of local $H_{rad}^1$ solutions near $r=0^+$
  6. Classification of positive local $H_{rad}^1$ solutions
  7. Uniqueness of the ground state
  8. Zeros of solutions to the linearization of \ref{['12']}
  9. Uniqueness of the ground state solution
  10. Construction of Local Stable Manifold
  11. Equation for perturbations
  12. Preliminary estimates
  13. Strichartz estimate of the linear flow
  14. Solving equation \ref{['1102']}
  15. Rewrite (NLS$_a$) near the standing wave solution
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $d\ge 3$, $a\in (-(\tfrac{d-2}{2})^2,0)$, $0<p<\frac{4}{d-2}$, $\beta=\sqrt{(d-2)^2+4a}$. Then there exists a unique radial positive solution $Q\in H^1(\mathbb R^d)$ to 2. Moreover, $Q(x)$ satisfies 1) $Q(x)\in C^{\infty}(\mathbb R^d \setminus \{0\})$ and the radial derivative $Q_r(r)<0$. 2) The 3) There exists $c_0>0$ such that 4) Let $C_{GN}$ be the sharp constant in the Gagliardo-Nirenberg

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • ...and 48 more