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

Abstract

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 Sn. In order to study this we first employ the Mobius transform to transform the above Hardy-Hénon type system on the sphere Sn into a higher order elliptic system on Rn. Then, we show that every positive solution of the higher order elliptic system on Rn is a solution to the associated integral system on Rn 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 Rn. Finally, together with the symmetry of the sphere Sn, 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.