Partial differential equations · Result 370

The Lane–Emden and Hénon–Lane–Emden conjectures

1 paperLean

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 \(A,B\gt -2\).

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 1

The Subcritical Hénon–Lane–Emden Conjecture

OpenAI · September 24, 2026

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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@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}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

The Subcritical Hénon–Lane–Emden Conjecture

Scope

The formalized result proves nonexistence of positive entire solutions to the subcritical Hénon–Lane–Emden system. For \(n\ge2\), \(p,q>0\), and real \(A,B\) with \((n+A)/(p+1)+(n+B)/(q+1)>n-2\), there are no positive functions, continuous everywhere and \(C^2\) off the origin, satisfying \(-\Delta u=|x|^A v^p\) and \(-\Delta v=|x|^B u^q\) away from the origin. It includes the globally \(C^2\) unweighted case for \(n\ge3\). The critical equality case is not included.

Comparator links

Result	Comparator statement
Subcritical Hénon–Lane–Emden nonexistence	HenonEmden.lean

Scope note on GitHub →