Global existence, blow-up and stability of standing waves for the Schrödinger-Choquard equation with harmonic potential

Meixia Cai; Hui Jian; Min Gong; Meixia Cai; Hui Jian; Min Gong

doi:10.3934/math.2024027

AIMS Mathematics

2024, Volume 9, Issue 1: 495-520. doi: 10.3934/math.2024027

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Global existence, blow-up and stability of standing waves for the Schrödinger-Choquard equation with harmonic potential

College of Science, East China Jiaotong University, Nanchang 330013, China

Received: 09 September 2023 Revised: 05 November 2023 Accepted: 15 November 2023 Published: 30 November 2023
MSC : 35A01, 35B33, 35Q40, 35B44, 35Q55

In this article, we conduct a comprehensive investigation into the global existence, blow-up and stability of standing waves for a $ L^{2} $-critical Schrödinger-Choquard equation with harmonic potential. First, by taking advantage of the ground-state solutions and scaling techniques, we obtain some criteria for the global existence and blow-up of the solutions. Second, in terms of the refined compactness argument, scaling techniques and the variational characterization of the ground state solution to the Choquard equation with $ p_{2} = 1+\frac{2+\alpha}{N} $, we explore the limiting dynamics of blow-up solutions to the $ L^{2} $-critical Choquard equation with $ L^{2} $-subcritical perturbation, including the $ L^{2} $-mass concentration and blow-up rate. Finally, the orbital stability of standing waves is investigated in the presence of $ L^{2} $-subcritical perturbation, focusing $ L^{2} $-critical perturbation and defocusing $ L^{2} $-supercritical perturbation by using variational methods. Our results supplement the conclusions of some known works.
- Schrödinger-Choquard equation,
- harmonic potential,
- global existence,
- blow-up,
- orbital stability
Citation: Meixia Cai, Hui Jian, Min Gong. Global existence, blow-up and stability of standing waves for the Schrödinger-Choquard equation with harmonic potential[J]. AIMS Mathematics, 2024, 9(1): 495-520. doi: 10.3934/math.2024027

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Abstract

In this article, we conduct a comprehensive investigation into the global existence, blow-up and stability of standing waves for a $ L^{2} $-critical Schrödinger-Choquard equation with harmonic potential. First, by taking advantage of the ground-state solutions and scaling techniques, we obtain some criteria for the global existence and blow-up of the solutions. Second, in terms of the refined compactness argument, scaling techniques and the variational characterization of the ground state solution to the Choquard equation with $ p_{2} = 1+\frac{2+\alpha}{N} $, we explore the limiting dynamics of blow-up solutions to the $ L^{2} $-critical Choquard equation with $ L^{2} $-subcritical perturbation, including the $ L^{2} $-mass concentration and blow-up rate. Finally, the orbital stability of standing waves is investigated in the presence of $ L^{2} $-subcritical perturbation, focusing $ L^{2} $-critical perturbation and defocusing $ L^{2} $-supercritical perturbation by using variational methods. Our results supplement the conclusions of some known works.

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