Positive solutions to the nonlinear Choquard equation: symmetry, uniqueness, and nondegeneracy
Abstract
We study the symmetry, uniqueness, and nondegeneracy of positive solutions to the Choquard equation −Δu+u=(Iα*up)up−1 in ℝN, which have long remained open problems. For N≥3, α∈(0,N), and (N+α)/N<p<(N+α)/(N−2),we prove that every positive H1-solution is radially symmetric and strictly decreasing in the radial variable. The symmetry proof relies on the moving-plane method, where the sublinear singularity is controlled by analyzing the far-field behavior of the nonlocal term. Uniqueness and nondegeneracy are established through a functional framework hinging on a nonnegative joint form and a Schur complement argument.Radial nondegeneracy and Morse index one are obtained by ruling out the spectral interval (0,p−1] via a strict cumulative Riesz identity, which leads to uniqueness through a Nehari two-well argument. Nondegeneracy in the full space H1(ℝN) follows from the spherical-harmonic decomposition of the linearized form. These results hold for the full range of p when α=2, and for p≥2 when α∈(2,N). The cases α∈(0,2) and α∈(2,N) with p<2 remain open.
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BibTeX
@misc{xia2026positive,
title = {{Positive solutions to the nonlinear Choquard equation: symmetry, uniqueness, and nondegeneracy}},
author = {Jiankang Xia and Chengxiang Zhang and Xu Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05138},
eprint = {2611.05138},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05138},
}
RIS
TY - UNPB
TI - Positive solutions to the nonlinear Choquard equation: symmetry, uniqueness, and nondegeneracy
AU - Jiankang Xia
AU - Chengxiang Zhang
AU - Xu Zhang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05138
UR - https://arxiv.org/abs/2611.05138
AB - We study the symmetry, uniqueness, and nondegeneracy of positive solutions to the Choquard equation −Δu+u=(I_α*u^p)u^p−1 in ℝ^N, which have long remained open problems. For N≥3, α∈(0,N), and (N+α)/N<p<(N+α)/(N−2),we prove that every positive H^1-solution is radially symmetric and strictly decreasing in the radial variable. The symmetry proof relies on the moving-plane method, where the sublinear singularity is controlled by analyzing the far-field behavior of the nonlocal term. Uniqueness and nondegeneracy are established through a functional framework hinging on a nonnegative joint form and a Schur complement argument.Radial nondegeneracy and Morse index one are obtained by ruling out the spectral interval (0,p−1] via a strict cumulative Riesz identity, which leads to uniqueness through a Nehari two-well argument. Nondegeneracy in the full space H^1(ℝ^N) follows from the spherical-harmonic decomposition of the linearized form. These results hold for the full range of p when α=2, and for p≥2 when α∈(2,N). The cases α∈(0,2) and α∈(2,N) with p<2 remain open.
ER -