Liouville-type theorems for polyharmonic systems in [formula omitted

Abstract

In this note, we consider the polyharmonic system ( − Δ ) m u = v α , ( − Δ ) m v = u β in R N with N > 2 m and α ⩾ 1 , β ⩾ 1 , where ( − Δ ) m is the polyharmonic operator. For 1 / ( α + 1 ) + 1 / ( β + 1 ) > ( N − 2 m ) / N , we prove the non-existence of non-negative, radial, smooth solutions. For 1 < α , β < ( N + 2 m ) / ( N − 2 m ) , we show the non-existence of non-negative smooth solutions. In addition, for either ( N − 2 m ) β < N α + 2 m or ( N − 2 m ) α < N β + 2 m with α , β > 1 , we show the non-existence of non-negative smooth solutions for polyharmonic system of inequalities ( − Δ ) m u ⩾ v α , ( − Δ ) m v ⩾ u β . More general, we can prove that all the above results hold for the system ( − Δ ) m u = v α , ( − Δ ) n v = u β in R N with N > max { 2 m , 2 n } and α ⩾ 1 , β ⩾ 1 .