A Liouville-type theorem for higher order elliptic systems

We prove there are no positive radial solutions for higher order elliptic system { ( − Δ ) m u = v p ( − Δ ) m v = u q in R N if 1 p + 1 + 1 q + 1 > 1 − 2 m N . We also show there are no positive solutions to the system under the additional assumption that max ( 2 ( p + 1 ) p q − 1 , 2 ( q + 1 ) p q − 1 ) ⩾ N − 2 m m . The proof in the radial case uses Rellichʼs identity and the proof in the general case relies on growth estimates of the spherical average of the solution.