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Positive solutions of Yamabe-type equations with general potentials

  • Hongyu Chen
  • Li Ma
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Abstract

Let (Mn,g) be a smooth closed connected Riemannian manifold of dimension n≥3, and let ψ∈ C∞(M,ℝ). We prove that the equation −(4(n−1))/(n−2)Δg u+ψu=u(n+2)/(n−2) has a smooth positive solution if and only if its linear part has positive first eigenvalue. In dimensions n≥6, the proof combines an energy estimate under exact local mass constraints with an intersection argument for finite cycles of atomic measures; dimensions 3≤ n≤5 follow from the theorem of Bahri and Brezis.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026positive,
  title = {{Positive solutions of Yamabe-type equations with general potentials}},
  author = {Hongyu Chen and Li Ma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34417},
  eprint = {2610.34417},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34417},
}
RIS
TY  - UNPB
TI  - Positive solutions of Yamabe-type equations with general potentials
AU  - Hongyu Chen
AU  - Li Ma
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34417
UR  - https://arxiv.org/abs/2610.34417
AB  - Let (M^n,g) be a smooth closed connected Riemannian manifold of dimension n≥3, and let ψ∈ C^∞(M,ℝ). We prove that the equation −(4(n−1))/(n−2)Δ_g u+ψu=u^(n+2)/(n−2) has a smooth positive solution if and only if its linear part has positive first eigenvalue. In dimensions n≥6, the proof combines an energy estimate under exact local mass constraints with an intersection argument for finite cycles of atomic measures; dimensions 3≤ n≤5 follow from the theorem of Bahri and Brezis.
ER  -