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Symmetry, uniqueness and nondegeneracy of positive solutions of the Choquard equation
Abstract
We study positive solutions Q∈ H1(ℝN) of −ΔQ+Q=(Iα*Qp)Qp−1, where Iα is the Riesz potential of order α, N≥1, 0<α<N, p>1+α/N, and (N−2)p<N+α. We prove that every such solution is radially symmetric and strictly decreasing about some point. For p≥2, we establish uniqueness up to translations and show that the real linearized operator has Morse index one and kernel spanned by ∂1Q,…,∂NQ. The symmetry proof uses the method of moving planes. Uniqueness follows from nondegeneracy and compactness through a continuation argument.
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BibTeX
@misc{fan2026symmetry,
title = {{Symmetry, uniqueness and nondegeneracy of positive solutions of the Choquard equation}},
author = {Song Fan and Gui-Dong Li and Jianjun Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02980},
eprint = {2611.02980},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02980},
}
RIS
TY - UNPB
TI - Symmetry, uniqueness and nondegeneracy of positive solutions of the Choquard equation
AU - Song Fan
AU - Gui-Dong Li
AU - Jianjun Zhang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02980
UR - https://arxiv.org/abs/2611.02980
AB - We study positive solutions Q∈ H^1(ℝ^N) of −ΔQ+Q=(I_α*Q^p)Q^p−1, where I_α is the Riesz potential of order α, N≥1, 0<α<N, p>1+α/N, and (N−2)p<N+α. We prove that every such solution is radially symmetric and strictly decreasing about some point. For p≥2, we establish uniqueness up to translations and show that the real linearized operator has Morse index one and kernel spanned by ∂_1Q,…,∂_NQ. The symmetry proof uses the method of moving planes. Uniqueness follows from nondegeneracy and compactness through a continuation argument.
ER -