Method of scaling spheres for integral and polyharmonic systems

Abstract

We establish a method of scaling spheres for the integral system [Formula presented] where 0<α,β<n, a>−α, b>−β and p,q>0. By using this method, we obtain a Liouville theorem for nonnegative solutions when [Formula presented], [Formula presented] and [Formula presented]. As an application, we derive a Liouville theorem for nonnegative solutions of the polyharmonic Hénon-Hardy system {(−Δ)mu(x)=|x|avp(x) in Rn,(−Δ)lv(x)=|x|buq(x) in Rn, where m and l are integers in [Formula presented].