The Lane-Emden conjecture for elliptic systems, with its Henon-weighted extension

The Lane-Emden system is $-\Delta u=v^p$ , $-\Delta v=u^q$ in $\mathbb R^n$ with $p,q>0$ . The Lane-Emden conjecture asserts that it has no positive classical entire solution in the subcritical region $\frac1{p+1}+\frac1{q+1}>\frac{n-2}n$ , with no symmetry or decay assumption. It was proved for $n\le2$ , for $n=3$ (Serrin-Zou; Polacik-Quittner-Souplet) and $n=4$ (Souplet), and in partial ranges for $n\ge5$ . Phan (2012, Conjecture C) posed the weighted Henon-Lane-Emden form $-\Delta u=|x|^Av^p$ , $-\Delta v=|x|^Bu^q$ for $A,B>-2$ , with subcritical region $\frac{n+A}{p+1}+\frac{n+B}{q+1}>n-2$ , and Huang and Zou stated it for all real weights with solutions continuous at the origin. Is there no positive entire solution throughout the subcritical region, in every dimension?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Elliptic systems; Liouville theorems
Posed by: Lane-Emden conjecture: Liouville literature of the 1990s (Mitidieri 1993, 1996; Serrin and Zou 1996); weighted form: Phan (2012), Conjecture C; Huang and Zou (2026), Conjecture B
Year posed: —
Years open: —
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 40 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Claims Theorem 1.1: for every $n\ge2$ , $p,q>0$ and $A,B\in\mathbb R$ with $\frac{n+A}{p+1}+\frac{n+B}{q+1}>n-2$ , the Henon-Lane-Emden system has no positive solution continuous at the origin and $C^2$ elsewhere, with no condition at infinity. Corollary 1.2 is the unweighted Lane-Emden conjecture; with Bidaut-Veron and Giacomini's radial existence theorem this gives Phan's Conjecture C in full for $n\ge3$ , $A,B>-2$ , and Huang-Zou's Conjecture B. It does NOT classify solutions on the critical hyperbola beyond the cited radial existence, and treats only positive (not sign-changing) solutions.

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. It combines a localized potential-kernel virial identity with an energy-relative interval-pressure estimate; the radial existence theorem of Bidaut-Veron and Giacomini used for the classification is credited to them.

Verification

No independent mathematician has checked this yet. Theorem 1.1 and Corollary 1.2 were read against the conjecture: for $n\ge2$ , $p,q>0$ , real $A,B$ in the weighted subcritical region, no positive solutions continuous on $\mathbb R^n$ and $C^2$ off the origin; the unweighted Lane-Emden conjecture is the case $A=B=0$ , $n\ge3$ . formalization.yaml lists ComparatorChallenges/HenonEmden.json, declaration OAI.HenonLaneEmden.main_nonexistence, file OAI/Analysis/HenonEmden/Main.lean. The statement was read here: for $n\ge2$ , $p,q>0$ , real $A,B$ with $(n+A)/(p+1)+(n+B)/(q+1)>n-2$ , there are no $u,v$ on Euclidean $n$ -space that are continuous, $C^2$ off the origin, strictly positive and satisfy both equations away from the origin. That is the headline nonexistence claim. The existence half of Phan's Conjecture C (via the cited radial theorem) is not formalised. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.

The Lane-Emden conjecture for elliptic systems, with its Henon-weighted extension

What was actually shown

What the AI did

Verification

Sources

Discussion