Orbital stability of positive standing waves to the logarithmic Choquard equations
Abstract
The paper concerns the orbital stability of positive standing waves for the logarithmic Choquard equation with a small local nonlinear perturbation. First, the uniqueness and nondegeneracy of the positive ground-state solution are established via bifurcation and compactness arguments. Then, by exploiting the spectral properties of the linearized operator around the positive ground-state solution in high-energy spaces, the orbital stability of the corresponding positive standing wave is proved. The main difficulty in establishing orbital stability stems from the fact that the logarithmic convolution kernel is both sign-changing and strongly singular, while the norm used to prove global existence is not translation invariant. Consequently, existing approaches to orbital stability for positive ground-state solutions of three-and higher-dimensional Choquard equations with Riesz-type kernels cannot be applied to the two-dimensional logarithmic Choquard equation considered here.
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BibTeX
@misc{liu2026orbital,
title = {{Orbital stability of positive standing waves to the logarithmic Choquard equations}},
author = {Zhisu Liu and Shu-Ming Sun and Weinian Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34200},
eprint = {2610.34200},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34200},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Orbital stability of positive standing waves to the logarithmic Choquard equations
AU - Zhisu Liu
AU - Shu-Ming Sun
AU - Weinian Zhang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34200
UR - https://arxiv.org/abs/2610.34200
AB - The paper concerns the orbital stability of positive standing waves for the logarithmic Choquard equation with a small local nonlinear perturbation. First, the uniqueness and nondegeneracy of the positive ground-state solution are established via bifurcation and compactness arguments. Then, by exploiting the spectral properties of the linearized operator around the positive ground-state solution in high-energy spaces, the orbital stability of the corresponding positive standing wave is proved. The main difficulty in establishing orbital stability stems from the fact that the logarithmic convolution kernel is both sign-changing and strongly singular, while the norm used to prove global existence is not translation invariant. Consequently, existing approaches to orbital stability for positive ground-state solutions of three-and higher-dimensional Choquard equations with Riesz-type kernels cannot be applied to the two-dimensional logarithmic Choquard equation considered here.
ER -