Method of scaling spheres for integral and polyharmonic systems

Abstract

We establish a method of scaling spheres for the integral system { u ( x ) = ∫ R n | y | a v p ( y ) | x − y | n − α d y , x ∈ R n , v ( x ) = ∫ R n | y | b u q ( y ) | x − y | n − β d y , x ∈ R n , 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 0 < p ≤ n + α + 2 a n − β , 0 < q ≤ n + β + 2 b n − α and ( p , q ) ≠ ( n + α + 2 a n − β , n + β + 2 b n − α ) . As an application, we derive a Liouville theorem for nonnegative solutions of the polyharmonic Hénon-Hardy system { ( − Δ ) m u ( x ) = | x | a v p ( x ) in R n , ( − Δ ) l v ( x ) = | x | b u q ( x ) in R n , where m and l are integers in ( 0 , n 2 ) .