The polyharmonic Hénon--Lane--Emden conjecture

Abstract

We study positive classical solutions of the polyharmonic Hénon--Lane--Emden system $(-\Delta {)}^{m}u=|x{|}^{\alpha }{v}^p$, $(-\Delta {)}^{m}v=|x{|}^{\beta }{u}^q$ in ${\mathbb{R}}^n$, where $m\ge 2$, $p,q\ge 1$, $pq>1$, and $\alpha ,\beta \ge 0$. We prove that there is no such solution whenever $\frac{n+\alpha }{p+1}+\frac{n+\beta }{q+1}>n-2m$, 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 $m\ge 2$: for these exponents and weights, a positive classical solution exists if and only if $\frac{n+\alpha }{p+1}+\frac{n+\beta }{q+1}\le n-2m$. The proof has two parts. We first show that positivity forces $(-\Delta {)}^{i}u>0$ and $(-\Delta {)}^{i}v>0$ for $1\le i\le m-1$. 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 $\mu \in (1,n)$, adapting a recent argument for the second-order case that combines a localized virial identity with an interval comparison along source layers.

arXiv commentThis research was initially deposited on Zenodo on October 9, 2026 (https://doi.org/10.5281/zenodo.23268558)