OpenAI · 370-01 · 2026-09-24 · Partial differential equations

The Subcritical Hénon–Lane–Emden Conjecture

preprints/The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026

Family 370. The subcritical Lane–Emden and Hénon–Lane–Emden conjectures.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For $n \geq 2$ , $p, q > 0$ and real $A, B$ , the system $- Δ u = ∣ x ∣^{A} v^{p}$ , $- Δ v = ∣ x ∣^{B} u^{q}$ has no positive entire solution when $(n + A) / (p + 1) + (n + B) / (q + 1) > n - 2$ , 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 $n \geq 3$ and $A, B > - 2$ .

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 $(n + A) / (p + 1) + (n + B) / (q + 1) > n - 2$ . Solutions need only be continuous at the origin and classical elsewhere, without a condition at infinity. For n ≥ 3 and $A, B > - 2$ , 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.

Preprint source and Lean scope

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This preprint is named in the Lean scope note for 370: The Lane–Emden and Hénon–Lane–Emden conjectures. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.

HenonEmden.lean · HenonEmden.lean

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