Nonexistence of subcritical Lane–Emden equilibria in higher dimensions
For $p, q > 0$ consider the Lane–Emden system $$-\Delta u = v^p, \qquad -\Delta v = u^q \qquad \text{on } \mathbb{R}^n.$$ The Lane–Emden conjecture asserts that it has no positive entire solution when $(p, q)$ lies below the critical Sobolev hyperbola, that is, when $\frac{1}{p+1} + \frac{1}{q+1} > 1 - \frac{2}{n}$. Souplet records that this had been proved for radial solutions and for $n \le 3$; he proved it for $n = 4$ and obtained a further region of nonexistence for $n \ge 5$. Li, Li and Wei proved it for $n \ge 5$ when $p, q \ge 1$ and $\frac{1}{p+1} + \frac{1}{q+1} \ge 1 - \frac{2}{n} + \frac{4}{n^2}$.
Problem. For every integer $n \ge 5$ and all exponents $p, q > 0$ with $$\frac{1}{p+1} + \frac{1}{q+1} > 1 - \frac{2}{n},$$ is it impossible to find strictly positive functions $u, v \in C^2(\mathbb{R}^n)$ satisfying this system? No radial symmetry, stability, decay, integrability or finite-energy assumption is imposed.
Source: The Lane–Emden conjecture, as stated by Souplet (2009) and by Li, Li and Wei (2025). Stated by AIM — Open Applied Problems for n ≥ 5.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, Lane–Emden systems, Liouville theorems
References
@article{souplet2009,
author = {Souplet, Philippe},
title = {The proof of the {L}ane--{E}mden conjecture in four space dimensions},
journal = {Adv. Math.},
volume = {221},
number = {5},
year = {2009},
pages = {1409--1427},
doi = {10.1016/j.aim.2009.02.014},
}
@misc{li2025,
author = {Li, Kui and Li, Mingxiang and Wei, Juncheng},
title = {On a new region for the {L}ane--{E}mden conjecture in higher dimensions},
year = {2025},
eprint = {2510.06613},
archivePrefix = {arXiv},
}
@misc{aim-lane-emden-subcritical-system,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Nonexistence of subcritical Lane–Emden equilibria in higher dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/350-lane-emden-subcritical-system.md},
note = {Entry 350 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove the subcritical Hénon--Lane--Emden conjecture: for every dimension $n\ge2$, 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 \(n\ge3\) and \(A,B>-2\), combining this result with the radial existence theorem of Bidaut-Véron and Giacomini proves …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00693, "Nonexistence of subcritical Lane–Emden equilibria in higher dimensions", clopen.solutions.