Liouville-type theorems for polyharmonic H\'enon-Lane-Emden system

TL;DR

This paper investigates Liouville-type theorems for polyharmonic Hénon-Lane-Emden systems, providing partial results in higher dimensions and supporting a conjecture about the nonexistence of positive solutions based on certain parameter conditions.

Contribution

The paper extends previous results by establishing partial nonexistence theorems for positive solutions in higher dimensions for the polyharmonic Hénon-Lane-Emden system.

Findings

Supports the conjecture for nonexistence of positive solutions when (N+a)/(p+1)+(N+b)/(q+1)>N-2m.

Provides partial results in dimensions N ≥ 2m+2.

Builds on Fazly's work for radial and low-dimensional solutions.

Abstract

We study Liouville-type theorem for polyharmonic H\'enon-Lane-Emden system $(-\Delta)^mu=|x|^av^p,\; (-\Delta)^mv=|x|^bu^q$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions should be true if and only if $(N+a)/(p+1)+$ $(N+b)/(q+1)>N-2m$. It is shown by Fazly [6] that the conjecture holds for radial solutions in all dimensions and for classical solutions in dimension $N\leq 2m+1$. We here give some partial results in dimension $N\geq 2m+2$.