Liouville-type theorems for polyharmonic systems in

In this note, we consider the polyharmonic system (−Δ)mu=vα, (−Δ)mv=uβ in with N>2m and α 1, β 1, where (−Δ)m 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 (−Δ)mu vα, (−Δ)mv uβ. More general, we can prove that all the above results hold for the system (−Δ)mu=vα,(−Δ)nv=uβ in with N>max{2m,2n} and α 1, β 1.