Liouville-type theorems for polyharmonic systems in RN

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