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Liouville Theorem for the Constant Q/R-Curvature Equation

  • Han Lu
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Abstract

We study positive entire solutions of the flat constant Q/R-curvature equation in dimension n≥5. We prove that the weak superharmonicity condition −Δu≥0 already forces every nonconstant solution to be admissible. With this automatic admissibility, we classify all such solutions as positive constants or standard bubbles. We also show that the weak superharmonicity condition is structurally necessary. Once it is removed, there is a one-parameter family of positive radial entire solutions near every positive constant. These solutions converge to a positive constant at infinity, while both −Δu and −Δ(u(n−2)/(n−4)) change sign infinitely many times.

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Cite

BibTeXRIS
BibTeX
@misc{lu2026liouville,
  title = {{Liouville Theorem for the Constant $Q/R$-Curvature Equation}},
  author = {Han Lu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07256},
  eprint = {2609.07256},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07256},
}
RIS
TY  - UNPB
TI  - Liouville Theorem for the Constant Q/R-Curvature Equation
AU  - Han Lu
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07256
UR  - https://arxiv.org/abs/2609.07256
AB  - We study positive entire solutions of the flat constant Q/R-curvature equation in dimension n≥5. We prove that the weak superharmonicity condition −Δu≥0 already forces every nonconstant solution to be admissible. With this automatic admissibility, we classify all such solutions as positive constants or standard bubbles. We also show that the weak superharmonicity condition is structurally necessary. Once it is removed, there is a one-parameter family of positive radial entire solutions near every positive constant. These solutions converge to a positive constant at infinity, while both −Δu and −Δ(u^(n−2)/(n−4)) change sign infinitely many times.
ER  -