Liouville-type theorems for higher-order Lane-Emden system in exterior domains

In this paper, we are mainly concerned with the following system in an exterior domains: (-Δ)mu = vp,u ≥ 0 inRN\B¯,(-Δ)mv = uq,v ≥ 0 inRN\B¯,Δiu = 0, Δiv = 0,i = 0,...,m - 1on∂B, where N ≥ 2m, m ≥ 1 is an integer, B = B1(0) = {x RN||x| < 1}, and (-Δ)m is the polyharmonic operator. We prove the nonexistence of positive solutions to the above system for 1 < p,q < N+2m N-2m if N > 2m, and 1 < p,q < +∞ if N = 2m. The novelty of the paper is that we do not ask u,v satisfy any symmetry and asymptotic conditions at infinity. By proving the superharmonic properties of the solutions, we establish the equivalence between systems of partial differential equations (PDEs) and integral equations (IEs), then the method of scaling sphere in integral form can be applied to prove the nonexistence of the solutions.