Liouville-type theorem for higher order Hardy-Hénon type systems on the sphere

In this paper, we study Liouville type theorems for the positive solutions to the following higher order Hardy-Hénon type system involving the conformal GJMS operator on the sphere $S^n$. In order to study this we first employ the Mobius transform to transform the above Hardy-Hénon type system on the sphere $S^n$ into a higher order elliptic system on $R^n$. Then, we show that every positive solution of the higher order elliptic system on $R^n$ is a solution to the associated integral system on $R^n$ by using polyharmonic average and iteration arguments. We use the method of moving planes in integral form to prove that there are no positive solutions for the integral system on $R^n$. Finally, together with the symmetry of the sphere $S^n$, we obtain the Liouville type theorem of the higher order Hardy-Hénon type system involving the GJMS operator on the sphere. The results of this paper are also new even for the Lane-Emden system on the sphere.