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Sharp Exponent of Stable Standing Waves for the Perturbated Hartree Equation

Guoyi Fu, Shanshan Fu, Xiaoguang Li, Jian Zhang, Shihui Zhu

Abstract

This paper is concerned with the stability of standing waves for the mass-critical Hartree equation with a focusing perturbation by the variational method. The profile decomposition theory is employed to prove the attainability of the cross constrained variational problem, and then the comparison of two cross constrained variational problems is derived. The sharp criteria of blowup, the orbital stability, and strong instability of standing waves without any frequency constraint are obtained. This improves the cross constrained variational argument proposed by Zhang (2005).

Sharp Exponent of Stable Standing Waves for the Perturbated Hartree Equation

Abstract

This paper is concerned with the stability of standing waves for the mass-critical Hartree equation with a focusing perturbation by the variational method. The profile decomposition theory is employed to prove the attainability of the cross constrained variational problem, and then the comparison of two cross constrained variational problems is derived. The sharp criteria of blowup, the orbital stability, and strong instability of standing waves without any frequency constraint are obtained. This improves the cross constrained variational argument proposed by Zhang (2005).

Paper Structure

This paper contains 11 sections, 17 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Variational characterization of ground states
  4. Main results
  5. Outline of this paper
  6. Preliminaries
  7. Variarional Characterization
  8. Sharp Criteria of Blowup
  9. Stability Result for Standing Wave
  10. Orbital Stability
  11. Strong Instability

Key Result

Proposition 1.1

Let $D\ge 3$ and $\omega>0$. If $1+\frac{4}{D}\le p<1+\frac{4}{D-2}$, then there exists $u\in \mathfrak{N}$ such that $\mathcal{L}[u]=d_{\mathfrak{N}}$, and $u$ is the ground state solution of Eq. elliptic.

Theorems & Definitions (40)

  • Proposition 1.1
  • Remark 1.2
  • Proposition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 30 more