The Lane–Emden and Hénon–Lane–Emden conjectures

What OpenAI claims

Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 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 ≥ 3 and A,B > -2. (Lean)

Paper

1. The Subcritical Hénon–Lane–Emden Conjecture

   24 September 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.