Uniqueness and nondegeneracy for a critical Choquard equation with Hardy potential

Abstract

We study the critical Choquard equation −Δu−μ|x|⁻²u =(|·|^2−N*|u|^p)|u|^p−2u in ℝ^N, where |·|^2−N is the Newton kernel, p=(N+2)/(N−2), and 0<μ<((N−2)²)/4. For every N≥3, we prove that positive solutions in D^1,2(ℝ^N) are radially symmetric about the origin, unique up to dilation, and nondegenerate. Our proof uses normalized logarithmic derivatives to obtain a cooperative system. The resulting monotonicity principle underlies both the uniqueness and nondegeneracy arguments, and remains effective in the subquadratic range 1<p<2, corresponding to N≥7. Consequently, the linearized quadratic form has Morse index one and the sharp inequality admits a global quadratic remainder estimate. The best constant is strictly concave in the Hardy parameter on [0,((N−2)²)/4), is of class C² with strictly negative second derivative in the interior, and satisfies two-sided bounds that agree to first order at μ=0.

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BibTeX RIS

BibTeX

@misc{gu2026uniqueness,
  title = {{Uniqueness and nondegeneracy for a critical Choquard equation with Hardy potential}},
  author = {Guangze Gu and Zhipeng Yang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08146},
  eprint = {2611.08146},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08146},
}

RIS

TY  - UNPB
TI  - Uniqueness and nondegeneracy for a critical Choquard equation with Hardy potential
AU  - Guangze Gu
AU  - Zhipeng Yang
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08146
UR  - https://arxiv.org/abs/2611.08146
AB  - We study the critical Choquard equation −Δu−μ|x|^−2u =(|·|^2−N*|u|^p)|u|^p−2u in ℝ^N, where |·|^2−N is the Newton kernel, p=(N+2)/(N−2), and 0<μ<((N−2)^2)/4. For every N≥3, we prove that positive solutions in D^1,2(ℝ^N) are radially symmetric about the origin, unique up to dilation, and nondegenerate. Our proof uses normalized logarithmic derivatives to obtain a cooperative system. The resulting monotonicity principle underlies both the uniqueness and nondegeneracy arguments, and remains effective in the subquadratic range 1<p<2, corresponding to N≥7. Consequently, the linearized quadratic form has Morse index one and the sharp inequality admits a global quadratic remainder estimate. The best constant is strictly concave in the Hardy parameter on [0,((N−2)^2)/4), is of class C^2 with strictly negative second derivative in the interior, and satisfies two-sided bounds that agree to first order at μ=0.
ER  -

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