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The Subcritical Hénon–Lane–Emden Conjecture
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Abstract
We prove the subcritical Hénon–Lane–Emden conjecture: for every dimension n ≥ 2, positive powers p, q, and real weights A, B, the system has no strictly positive entire solution when $(n+A)/(p+1)+(n+B)/(q+1)\gt n-2$. Solutions need only be continuous at the origin and classical elsewhere, without a condition at infinity. For n ≥ 3 and $A,B\gt -2$, combining this result with the radial existence theorem of Bidaut-Véron and Giacomini proves Phan's Conjecture C in full: a positive radial entire solution exists whenever the strict inequality fails.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Linfeng Mei, Juncheng Wei, A proof of the fractional Lane–Emden conjecture (arXiv:2610.10188, 2026-10-07). Proves the fractional analogue ((−Δ)^s, 0<s<1) of the unweighted subcritical Liouville theorem, citing this preprint as reference [44] and adapting its localized potential-kernel virial and source-layer interval-pressure method.
- Kui Li, Haoyang Lu, Zhitao Zhang, The fractional Hénon–Lane–Emden conjecture and some critical nonexistence phenomena (arXiv:2610.22802v1, 2026-10-16). Builds on OpenAI's and Mei–Wei's kernel and interval comparisons to establish weighted fractional subcritical nonexistence and study critical-hyperbola phenomena for orders below one half, with AI assistance acknowledged for parts of the subcritical proof.
- Yuxia Guo, Yichen Hu, Shaolong Peng, Tingfeng Yuan, Classification of positive solutions to the critical Lane-Emden system (arXiv:2610.27875, 2026-10-20). Classifies positive solutions on the critical hyperbola without decay or energy assumptions, citing this paper ("OpenAI [27]") and adapting its "source-layer interval comparison method" to the critical case.
- mingfeng87 (GitHub), Lane–Emden-Type Conjecture (GitHub repository, proofs dated 2026-10-08). Draft candidate proofs for weighted fractional and weighted polyharmonic Lane–Emden systems adapting "the localized source-layer pressure method originally developed for second-order systems"; the README does not cite this paper and marks the proofs as not yet verified. [AI-generated per source]
Formalization
- OpenAI math release, The Lane–Emden and Hénon–Lane–Emden conjectures: Lean scope notes (OpenAI Math Release, 2026-10-06). States that the Lean development covers subcritical nonexistence (Theorem 1.1) for positive origin-continuous solutions with real weights, including the globally $C^2$ unweighted case for $n\ge 3$, with comparator statement
HenonEmden.lean; the critical equality case is excluded. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Haoyang Lu, Zhitao Zhang, New result on Lane-Emden conjecture with exponents in a disk (arXiv:2610.09526, 2026-10-07). An independent partial result on the unweighted Lane–Emden conjecture (Corollary 1.2 of this paper) for exponents in a disk-shaped region; its bibliography does not mention this paper.
Digests and expositions
- Latent Seminar, The weighted subcritical Lane–Emden system (2026-10-06). Explains the claimed theorem, scaling, localized virial identity, and weighted interval-pressure estimates, while stating that it is not an independent verification of the complete proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 370
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026,
author = {{OpenAI}},
title = {{The Subcritical H{\'e}non--Lane--Emden Conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026/paper.pdf}{OAI:The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/370.md. The Lean statements have not been checked by people against the paper's claims.
Family 370: The Lane–Emden and Hénon–Lane–Emden conjectures
Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, $p,q\gt 0$ and real A, B, the system $-\Delta u=|x|^A v^p$, $-\Delta v=|x|^B u^q$ has no positive entire solution when $(n+A)/(p+1)+(n+B)/(q+1)\gt n-2$, with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and $A,B\gt -2$.
This is the only paper in this family.