Result 370, Partial differential equations

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 $A,B\gt -2$ .

Lean formalization Proof

The bigger picture

Why it matters

Can two positive functions sustain one another through coupled spatial equations across an infinite space? The manuscript reports a sharp obstruction, identifying a range where no such pair can exist, even without symmetry or restrictions far away.

What changes?

In dimension n >= 2, take p,q > 0 and real A,B. The negative Laplacians (spatial curvatures) of u and v equal v to power p and u to power q, weighted by distance from the origin to powers A and B, respectively. The manuscript rules out everywhere-positive solutions when (n+A)/(p+1) + (n+B)/(q+1) > n-2. Solutions need only be continuous at the origin and classical (twice continuously differentiable) elsewhere, without symmetry or growth assumptions.

What does that help mathematicians do?

For n >= 3 and A,B > -2, combining this claim with the known radial existence theorem gives an exact dividing line: a positive entire solution exists precisely when the displayed strict inequality fails, including equality. The existing solutions can be radial, meaning they depend only on distance from the origin. Researchers could therefore rule out nonradial solutions as a way around the obstruction in the excluded range.

Are there practical applications?

The immediate value is foundational: this would classify when these nonlinear equations can support positive solutions throughout space. Setting A = B = 0 recovers the unweighted Lane–Emden problem; other weights describe coupling that varies with distance. The reported advance concerns existence and impossibility, not a numerical solver or a demonstrated practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Subcritical Hénon–Lane–Emden Conjecture

September 24, 2026 20 pages Main result formalized in Lean

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.

PDF LaTeX source

Cite (BibTeX)

@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's note on what the formalization covers, from lean/docs/370.md.

The Lane–Emden and Hénon–Lane–Emden conjectures

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

At a glance

Subject: Partial differential equations
Manuscripts: 1
Pages: 20
Dated: Sep 24, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Subcritical Lane–Emden conjectureProved
Subcritical Hénon–Lane–Emden conjectureProved
Phan's Conjecture C for $n \ge 3$ and $A,B\gt -2$ Proved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

Most similar results

375, Partial differential equationsDe Giorgi's conjecture in dimension eight
350, Differential geometryYau’s nodal bounds: surfaces and higher dimensions
369, Partial differential equationsThe hot spots conjecture for simply connected planar domains
368, Partial differential equationsThe three-dimensional Ball–Evans approximation problem
080, Real and complex analysisThe exact Sobolev endpoint for Schrödinger convergence
147, Dynamical systems and ergodic theoryThe near-boundary Birkhoff conjecture

By similarity of summaries and abstracts.

On the map

This result, outlined, and its most similar results among all 372. Open the full map.