A proof of the fractional Lane--Emden conjecture
Abstract
Let n≥2, 0<s<1, and p,q>0. We prove a Liouville theorem for the fractional Lane--Emden system (−Δ)s u=vp, (−Δ)s v=uq, u, v>0 in Rn. More precisely, we show that the system does not have a strictly positive entire solution whenever 1/(p+1)+1/(q+1)>(n−2s)/n. No radial symmetry, global boundedness, finite-energy assumption, or prescribed decay at infinity is required. Our proof adapts the localized potential-kernel virial and source-layer interval-pressure method recently introduced for the classical Hénon--Lane--Emden system. The basic reason the method also works in the fractional setting is simple: the Riesz kernel associated with (−Δ)s has the same logarithmic derivative structure as the Newton kernel, with n−2 replaced by n−2s. However, two genuinely nonlocal points require special treatments. When 2s≤1, the individual kernel-gradient terms are not locally absolutely integrable, so the two equations must be combined before the diagonal limit is taken. When 2s>n−1, the kernel restricted to a line is not integrable at infinity, and the interval comparison must therefore be formulated using nonnegative extended integrals. After these issues are resolved, a localized virial identity, a one-dimensional interval-pair inequality, and an averaged source-layer estimate yield a universal local energy bound. The subcritical scaling then forces the same energy to diverge, giving the desired contradiction.
Related
- The fractional Hénon-Lane-Emden conjecture and some critical nonexistence phenomena
- The Serrin exponent for Lane-Emden inequalities driven by sums of fractional p-Laplacians
- Classification of Anti-symmetric Solutions to Fractional Lane-Emden System in Borderline Case
- New result on Lane-Emden conjecture with exponents in a disk
- The polyharmonic Hénon--Lane--Emden conjecture
BibTeX
@misc{mei2026proof,
title = {{A proof of the fractional Lane--Emden conjecture}},
author = {Linfeng Mei and Juncheng Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10188},
eprint = {2610.10188},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10188},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A proof of the fractional Lane--Emden conjecture
AU - Linfeng Mei
AU - Juncheng Wei
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10188
UR - https://arxiv.org/abs/2610.10188
AB - Let n≥2, 0<s<1, and p,q>0. We prove a Liouville theorem for the fractional Lane--Emden system (−Δ)^s u=v^p, (−Δ)^s v=u^q, u, v>0 in R^n. More precisely, we show that the system does not have a strictly positive entire solution whenever 1/(p+1)+1/(q+1)>(n−2s)/n. No radial symmetry, global boundedness, finite-energy assumption, or prescribed decay at infinity is required. Our proof adapts the localized potential-kernel virial and source-layer interval-pressure method recently introduced for the classical Hénon--Lane--Emden system. The basic reason the method also works in the fractional setting is simple: the Riesz kernel associated with (−Δ)^s has the same logarithmic derivative structure as the Newton kernel, with n−2 replaced by n−2s. However, two genuinely nonlocal points require special treatments. When 2s≤1, the individual kernel-gradient terms are not locally absolutely integrable, so the two equations must be combined before the diagonal limit is taken. When 2s>n−1, the kernel restricted to a line is not integrable at infinity, and the interval comparison must therefore be formulated using nonnegative extended integrals. After these issues are resolved, a localized virial identity, a one-dimensional interval-pair inequality, and an averaged source-layer estimate yield a universal local energy bound. The subcritical scaling then forces the same energy to diverge, giving the desired contradiction.
ER -