A Liouville type theorem for higher order Hardy–Hénon equation in Rn

In this paper, we consider the following higher order Hardy–Hénon type equations in RnRn:equation(1)(−Δ)mu(x)=|x|aup(x),x∈Rn in subcritical cases with a>0a>0, and in particular, we focus on the non-existence of positive solutions.First, under some very mild growth conditions, we show that problem (1) is equivalent to the integral equationequation(2)u(x)=∫RnG(x,y)|y|aup(y)dy where G(x,y)G(x,y) is the Green's function associated with (−Δ)m(−Δ)m in RnRn.Then by using the method of moving planes in integral forms, we prove that there is no positive solution for integral equation (2) in subcritical cases nn−2m