Liouville-type theorems for polyharmonic Hénon-Lane-Emden system

Abstract

We study Liouville-type theorem for polyharmonic Hénon-Lane-Emden system $(-\Delta {)}^{m}u=|x{|}^a{v}^p,(-\Delta {)}^{m}v=|x{|}^b{u}^q$ when $m,p,q\ge 1,pq\ne 1$, and $a,b\ge 0$. 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 $N\le 2m+1$. We here give some partial results in dimension $N\ge 2m+2$.