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PreprintarXiv licencearXiv

Nonexistence of Stable Solutions to Lane--Emden Equations on Graphs with Polynomial Volume Growth

  • Weiqi Guan
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Abstract

We study stable solutions of the Lane--Emden equation −Δu=|u|p−1u on graphs with polynomial volume growth. Due to the lack of a chain rule in the discrete setting, we establish suitable algebraic inequalities which serve as substitutes for the usual differential identities. Combining these inequalities with the stability condition and cutoff arguments, we obtain nonexistence results for stable solutions in a range of exponents analogous to the Joseph--Lundgren range. As an application, we derive a lower bound for the Sobolev best constant in the discrete setting.

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Cite

BibTeXRIS
BibTeX
@misc{guan2026nonexistence,
  title = {{Nonexistence of Stable Solutions to Lane--Emden Equations on Graphs with Polynomial Volume Growth}},
  author = {Weiqi Guan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15275},
  eprint = {2610.15275},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15275},
}
RIS
TY  - UNPB
TI  - Nonexistence of Stable Solutions to Lane--Emden Equations on Graphs with Polynomial Volume Growth
AU  - Weiqi Guan
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15275
UR  - https://arxiv.org/abs/2610.15275
AB  - We study stable solutions of the Lane--Emden equation −Δu=|u|^p−1u on graphs with polynomial volume growth. Due to the lack of a chain rule in the discrete setting, we establish suitable algebraic inequalities which serve as substitutes for the usual differential identities. Combining these inequalities with the stability condition and cutoff arguments, we obtain nonexistence results for stable solutions in a range of exponents analogous to the Joseph--Lundgren range. As an application, we derive a lower bound for the Sobolev best constant in the discrete setting.
ER  -