Nonlinear elliptic partial differential equations · Liouville theorems · critical phenomena

Lane–Emden Conjecture

Collaboration beta

The conjecture says the coupled Lane–Emden system has no everywhere-positive classical solution on all of Euclidean space when the exponents lie strictly below the critical hyperbola.

p, q > 0 : ∄ u, v > 0 classical on ℝ^{n} : - Δ u = v^{p}, - Δ v = u^{q} when \frac{1}{p + 1} + \frac{1}{q + 1} > 1 - \frac{2}{n}

Known results and sources

Coupled positive fields and a narrowing endpoint layer introduce the analytic geometry of the Lane–Emden problem.

Research problem

Exact mathematical statement

For exponents $p,q>0$ and positive classical functions $u,v$ on $\mathbb R^n$ , consider

-\Delta u=v^p,\qquad -\Delta v=u^q,\qquad u,v>0\quad\text{in }\mathbb R^n.

The conjecture asserts that no such entire pair exists whenever

\frac{1}{p+1}+\frac{1}{q+1}>1-\frac{2}{n}.

After the known exclusions and swapping the two components, the unresolved working route may be restricted further to $n\ge5$ , $p>q>1$ , and $pq>1$ .

Problem infographic

Problem at a glance

Scientific explainer for the open Lane–Emden conjecture. Emerald and cobalt profiles represent positive functions u and v on R^n coupled by minus Laplacian u equals v to the p and minus Laplacian v equals u to the q. A reciprocal-exponent plane highlights the strict region 1/(p+1) + 1/(q+1) > 1 - 2/n and marks the equality line as the critical boundary, where critical ground states exist. The known cases n at most 4 and p equals q greater than 1 are separated from the still-open general case. — The Lane–Emden conjecture predicts that the coupled elliptic system has no positive classical entire solution in the strict subcritical region. The equality line is a genuine boundary where critical ground states exist; the full higher-dimensional asymmetric case remains open.

Current mathematical picture

Where work on Lane–Emden Conjecture stands

Partially resolved

Version 14 retains the earlier Lane–Emden research record while adding a source-reported balanced natural-scale core, a same-weight no-cascade estimate, the exact Bellman reconstruction, and a sharper finite-certificate work program. It also isolates the bounded-solution reduction as an external dependency that must be checked against its original source. These are provisional research-state additions; the full conjecture remains open.

Strongest supported footholdDimension-six direct hard cell reported

Version 14 reports bounded normalized nonexistence on the exact dimension-six hard cell as an unrefereed project theorem, while the arbitrary-entire statement retains an external reduction dependency.

Evidence posture · Reported result

Leading routeDirect dimension-six hard-cell certificate

Reproduce the exact direct certificate, balanced-core selection, same-weight maximal estimate, and bounded normalized contradiction on the certified dimension-six cell; separately audit the external reduction before extending the public statement to arbitrary entire solutions.

Route status · Active route

Useful failureHigh-ratio trace and recession classification

Uniform endpoint trace compactness is not a live route; v14 records endpoint recession classification as exact and reserves relaxed separation only for fallback continuation.

Route status · Eliminated route

Main reductionConditional contact–beta algebra delimited

After separation, the source reports contact-graph regularity as automatic from the two measure-adjoint equations; only applicable endpoint-flux installation remains explicit in the residual fallback route.

Evidence posture · Reported reduction

Completed special caseDimension-six direct hard-cell theorem

The v14 source reports nonexistence of bounded normalized positive solutions on the exact dimension-six hard cell as an exact project theorem; the corresponding arbitrary-entire statement still depends on the standard bounded-solution/doubling reduction.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeBuild the finite direct-certificate atlas

Cover the remaining compact hard region by rationally certified direct cells and connect them to the natural-core PDE theorem.

Task status · Work already reported in progress

Research-record correctionResearch-record correction

We removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Lane–Emden Conjecture in numbers

6.9kretained lines of mathematical investigation6,922 in the current working snapshot

Argument development: 5,699 · 82%
Explored or eliminated routes: 229 · 3%
Computational analysis: 192 · 3%
Open obligations: 511 · 7%
Definitions and setup: 291 · 4%

25selected mapped statements15routes investigated7reported milestones8open questions5contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 2, 2026

Source recordedAug 6, 2026

Recommended next task

Verify the bounded-solution reduction source

Check the exact literature statement used to pass from an arbitrary positive entire solution to a normalized bounded one.

Suggested move: Reopen the original Poláčik–Quittner–Souplet/Souplet statement and verify regularity, nontriviality, normalization, and compact-cell uniformity before any arbitrary-entire public theorem claim.

What would count as progress

A primary-source citation and exact scope audit cover all four reduction-interface conditions.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

28 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeDirect dimension-six hard-cell certificate

Route status · Active route

Active routeFinite direct-template atlas

Enlarge the current exact cell, add low-degree direct templates, and certify a finite rational atlas over the compact hard region.

Route status · Active route

Active routeNear-critical Jacobi/Bregman cover

Complete the retained local-inverse, nonlinear Bregman, second-order, and fourth-order package uniformly through the q down to 1 endpoint.

Route status · Active route

Active routeExact atlas/near-critical overlap

Prove explicit inequalities showing that the finite direct atlas and the near-critical neighborhood cover every remaining hard point.

Route status · Active route

Explored alternatives

Other routes

11 recorded

Narrowed routeInterior beta compactness

The corrected W¹,¹/BV argument is active and exact only on fixed ratio strips compactly contained in (0,T); its former endpoint-global reading is superseded.

Route status · Narrowed route

Eliminated routeHigh-ratio trace and recession classification

Uniform endpoint trace compactness is not a live route; v14 records endpoint recession classification as exact and reserves relaxed separation only for fallback continuation.

Route status · Eliminated route

Refuted routeUnconditional fixed raw-reserve padding

The x⁴−εx² witness rules out coefficient-small padding without finite, tangent, and vector-infinity contact checks.

Route status · Refuted route

Browse 8 more explored routes

Narrowed routeTangent-safe reserve control

The tangent-safe reserve program is no longer controlling on direct smooth-certificate cells and remains in the current research map only as fallback history.

Route status · Narrowed route

Route held in reserveResidual-cell relaxed separation

Use relaxed separation only on explicitly residual compact boxes where direct smooth templates fail.

Route status · Route held in reserve

Useful but insufficientFinite contact and endpoint-atom search

Local jets remove prescribed finite contacts and endpoint atoms, so repeating their classification is too weak for the noncompact endpoint and connected-contact problems.

Route status · Useful but insufficient

Not yet justifiedRoot-gap algebra without separation

The exact root-gap no-go remains useful only after a contact measure and admissible beta arc are obtained; it cannot provide them itself.

Route status · Not yet justified

Route held in reserveSame-homogeneity low-ratio interpolation

Retain low-ratio interpolation inside the paused residual continuation architecture, not as a primary compact-hard-region route.

Route status · Route held in reserve

Narrowed routeSpatial singular-current theorem

The v14 source records the singular-current chain rule as an exact project theorem; smooth direct cells need no additional spatial-current bridge.

Route status · Narrowed route

Route held in reserveResidual dimension-six continuation

The five-bridge continuation architecture is held in reserve outside cells covered by the direct hard-cell theorem and finite template atlas.

Route status · Route held in reserve

Narrowed routeFormer Mellin/root-gap global cover

The v12 global continuation cover is no longer the controlling compact-region strategy; v14 replaces it with a direct finite atlas plus a near-critical cover and exact overlap.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementWeighted entropy reaches the natural critical energy class

For a critical Jacobi pair with finite weighted entropy, the HLS/Riesz argument places φ and ψ and their Laplacians in the natural dual energy spaces. Turning this energy-class conclusion into geometric rigidity still requires the pinned critical nondegeneracy theorem in precisely that class.

Source-reported route statement · dependencies incomplete

Route statementCompact-asymmetry homogeneous Jacobi gap

The current work retains the compact-asymmetry homogeneous Jacobi gap as an exact project theorem under branch regularity, pending an independent checking of high-mode constants and fixed-core Gram bounds.

Source-reported route statement · dependencies incomplete

Route statementFourth-order homogeneous endpoint gap

The current work retains a fourth-order homogeneous endpoint gap as an exact but unrefereed project theorem, pending audits of the north-pole singular expansion and high-mode norm comparison.

Source-reported route statement · dependencies incomplete

Route statementBalanced natural-scale core

Uniformly on a compact exponent cell, every point has a nearby natural-scale ball on which M is comparable to its center value and the ratio t stays between a positive τ and T.

Source-reported route statement · dependencies incomplete

Route statementSame-weight no-cascade estimate

The weighted natural-scale mass is bounded by the same weighted full-density integral plus a cutoff error, without a cross-scale packing assertion.

Source-reported route statement · dependencies incomplete

Route statementDimension-six direct hard-cell theorem

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

8 featured tasks

Verify the bounded-solution reduction source

Check the exact literature statement used to pass from an arbitrary positive entire solution to a normalized bounded one.

Ready to work on

Complete the near-critical Jacobi/Bregman cover

Complete the uniform local inverse and nonlinear Bregman control through q=1, including the retained second-order and fourth-order near-critical package.

Suggested move: Prove the uniform local inverse and nonlinear Bregman estimate through q=1, then close the fourth-order endpoint.

Ready to work on

Residual endpoint-flux installation

After a positive separating contact measure exists, version 14 reports contact-graph regularity as automatic from the two measure-adjoint equations; only the applicable endpoint-flux installation remains as a residual fallback input.

Suggested move: Defer endpoint-flux installation to a documented residual fallback cell not covered by direct templates; do not reopen contact-graph regularity as an independent task.

Ready to work on

Residual-cell relaxed separation fallback

Retain relaxed first-loss separation only as a fallback for residual compact boxes where the direct smooth-template atlas fails.

Suggested move: Attempt the direct finite-template search first; reopen separation only for a precisely recorded residual box.

Ready to work on

Residual low-ratio continuation fallback

Keep low-ratio interpolation only inside the residual relaxed-continuation route rather than as a primary compact-hard-region task.

Suggested move: Use this only after a direct-template search leaves a documented residual compact box.

Ready to work on

Prove exact atlas/near-critical overlap

Give explicit parameter inequalities showing that the finite direct-certificate atlas and near-critical neighborhood cover every remaining hard point.

Suggested move: After the finite atlas and near-critical theorem are explicit, prove their exact overlap with parameter inequalities.

Blocked by the current route

Residual dimension-six continuation fallback

The former five-bridge dimension-six continuation remains a fallback outside the direct hard cell, not the controlling regional route.

Suggested move: First reproduce and enlarge the direct hard cell; use the continuation architecture only on a documented residual cell.

Blocked by the current route

Build the finite direct-certificate atlas

Cover the remaining compact hard region by rationally certified direct cells and connect them to the natural-core PDE theorem.

Suggested move: Enlarge the current rational hard cell by branch-and-bound, record the first failing polynomial on rejected boxes, and add low-degree templates where the fixed template stops.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026

Current statusPartially resolved

The conjecture is proved for dimensions n≤4 and for multiple subregions of the subcritical exponent range in higher dimensions, but the full higher-dimensional case n≥5 remains open.

[4][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025
PreprintLi, Li, and Wei proved nonexistence for n≥5, p,q≥1 when 1/(p+1)+1/(q+1)≥1−2/n+4/n², with a large-dimension variant replacing 4 by any fixed constant greater than 2.[3]
2018
PreprintLi and Zhang proved the conjecture throughout the subcritical region when min(p,q)≤1 and recovered the known condition pq≤1 or max(alpha,beta)>n−3.[2]
2009
Peer reviewedSouplet completed the conjecture in dimension four; together with earlier work this settles all dimensions n≤4.[4]
1993
PreprintMitidieri established the sharp radial-solution classification: positive radial solutions occur on or above the critical hyperbola and not below it.[1][2]

Mathematical neighborhood

Related results and reusable starting points

Current focusLane–Emden conjecture

Stronger or generalized formscalar Lane–Emden Liouville theorem

When the two components and exponents coincide, the system reduces to the scalar Lane–Emden equation, whose subcritical Liouville theorem was established by Gidas and Spruck.

[3]

Related problemSobolev critical hyperbola

The subcritical side is conjectured to have no positive entire solutions, while radial positive solutions exist on and above the critical hyperbola.

[1]

Related problemHénon–Lane–Emden conjecture

The weighted Hénon–Lane–Emden system is a distinct extension with its own weighted critical hyperbola and should not be conflated with the unweighted conjecture.

[5]

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Finite atlas and external reduction audit become explicitThe source separates the finite direct-cell program from the literature-dependent bounded-solution reduction that still needs a primary-source scope check.

Changed the research frontierLater mathematical revision

Version 14 frontier refinement

Balanced natural-scale core and no-cascade estimate addedVersion 14 reports the missing local ratio-balancing theorem, same-weight maximal estimate, and Bellman reconstruction needed by the direct regional route.

Changed the research frontierLater mathematical revision

Version 14 core analytic additions

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Claim connections clarified

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

7 mapped milestonesretained argument map

Browse all 7 mapped stages

stage 1Interior compactness corrected and completed
stage 2Unconditional raw padding refuted
stage 3Continuous separation and measure precursor
stage 4Contact–beta algebra delimited
stage 5Local source corridor made reproducible
stage 6Dimension-six arithmetic retained
stage 7Five analytic bridges define the regional frontier

Interior compactness corrected and completedVersion 12 supplies the missing integration-constant bridge and proves W¹,¹/BV compactness on fixed interior strips while removing the unsupported endpoint-global scope.