The polyharmonic Hénon--Lane--Emden conjecture
Abstract
We study positive classical solutions of the polyharmonic Hénon--Lane--Emden system (−Δ)m u=|x|αvp, (−Δ)m v=|x|βuq in ℝn, where m≥2, p,q≥1, pq>1, and α,β≥0. We prove that there is no such solution whenever (n+α)/(p+1)+(n+β)/(q+1)>n−2m, in every dimension and without any assumption on the behavior of the solutions at infinity. Combined with an existence result of Villavert, this proves the polyharmonic Hénon--Lane--Emden conjecture of Phan for m≥2: for these exponents and weights, a positive classical solution exists if and only if (n+α)/(p+1)+(n+β)/(q+1)≤ n−2m. The proof has two parts. We first show that positivity forces (−Δ)i u>0 and (−Δ)i v>0 for 1≤ i≤ m−1. For m even this needs pq>1, and explicit radial examples show that it fails when pq<1 and when p=q=1. The system is then equivalent to a weighted integral system with a Riesz kernel of order 2m. For such integral systems we prove the corresponding Liouville theorem for every Riesz order μ∈(1,n), adapting a recent argument for the second-order case that combines a localized virial identity with an interval comparison along source layers.
Related
- The fractional Hénon-Lane-Emden conjecture and some critical nonexistence phenomena
- New result on Lane-Emden conjecture with exponents in a disk
- A proof of the fractional Lane--Emden conjecture
- A Sharp Gradient Harnack Estimate for the Lane--Emden System
- Lane-Emden equation for the superposition of a local and nonlocal operator with the "wrong sign"
BibTeX
@misc{le2026polyharmonic,
title = {{The polyharmonic Hénon--Lane--Emden conjecture}},
author = {Phuong Le},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02448},
eprint = {2611.02448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02448},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The polyharmonic Hénon--Lane--Emden conjecture
AU - Phuong Le
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02448
UR - https://arxiv.org/abs/2611.02448
AB - We study positive classical solutions of the polyharmonic Hénon--Lane--Emden system (−Δ)^m u=|x|^αv^p, (−Δ)^m v=|x|^βu^q in ℝ^n, where m≥2, p,q≥1, pq>1, and α,β≥0. We prove that there is no such solution whenever (n+α)/(p+1)+(n+β)/(q+1)>n−2m, in every dimension and without any assumption on the behavior of the solutions at infinity. Combined with an existence result of Villavert, this proves the polyharmonic Hénon--Lane--Emden conjecture of Phan for m≥2: for these exponents and weights, a positive classical solution exists if and only if (n+α)/(p+1)+(n+β)/(q+1)≤ n−2m. The proof has two parts. We first show that positivity forces (−Δ)^i u>0 and (−Δ)^i v>0 for 1≤ i≤ m−1. For m even this needs pq>1, and explicit radial examples show that it fails when pq<1 and when p=q=1. The system is then equivalent to a weighted integral system with a Riesz kernel of order 2m. For such integral systems we prove the corresponding Liouville theorem for every Riesz order μ∈(1,n), adapting a recent argument for the second-order case that combines a localized virial identity with an interval comparison along source layers.
ER -