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PreprintCC BY 4.0arXiv

Uniqueness and Symmetry Breaking for Singular Q-curvature Equations in Rn

  • Haoyu Li
  • Tuoxin Li
  • Juncheng Wei
  • Wen Yang
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Abstract

For n≥3 and γ>−1, we study normal solutions of (−Δ)n/2u=(n−1)! |x|nγenu in Rn. We reformulate this problem on the cylinder via the Emden--Fowler transform. We first establish Kelvin symmetry for normal solutions. We then prove existence and uniqueness, up to dilation, of the radial normal solution and establish its radial nondegeneracy. Next, we carry out a detailed spectral analysis of the linearized operator and construct branches of nonradial normal solutions by the Crandall--Rabinowitz theorem. Finally, we prove that every normal solution is radial when γ>0 is sufficiently small.

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Cite

BibTeXRIS
BibTeX
@misc{li2026uniqueness,
  title = {{Uniqueness and Symmetry Breaking for Singular $Q$-curvature Equations in $\R^n$}},
  author = {Haoyu Li and Tuoxin Li and Juncheng Wei and Wen Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11056},
  eprint = {2610.11056},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11056},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Uniqueness and Symmetry Breaking for Singular Q-curvature Equations in R^n
AU  - Haoyu Li
AU  - Tuoxin Li
AU  - Juncheng Wei
AU  - Wen Yang
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11056
UR  - https://arxiv.org/abs/2610.11056
AB  - For n≥3 and γ>−1, we study normal solutions of (−Δ)^n/2u=(n−1)! |x|^nγe^nu in R^n. We reformulate this problem on the cylinder via the Emden--Fowler transform. We first establish Kelvin symmetry for normal solutions. We then prove existence and uniqueness, up to dilation, of the radial normal solution and establish its radial nondegeneracy. Next, we carry out a detailed spectral analysis of the linearized operator and construct branches of nonradial normal solutions by the Crandall--Rabinowitz theorem. Finally, we prove that every normal solution is radial when γ>0 is sufficiently small.
ER  -