A Liouville-type theorem for higher order elliptic systems

We prove there are no positive radial solutions for higher order elliptic system{(−Δ)mu=vp(−Δ)mv=uqinRN if 1p+1+1q+1>1−2mN. We also show there are no positive solutions to the system under the additional assumption that max(2(p+1)pq−1,2(q+1)pq−1)⩾N−2mm. 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.