A Liouville Type Theorem For Higher Order Hardy-Hénon Equations In R~n

In this paper, we consider the following higher order elliptical equations in Rn: in subcritical cases with a> 0, and in particular, we focus on the non-existence of positive solutions.First, under some very mild growth conditions, we show that problem (3) is equivalent to the integral equation where G(x, y) is the Green’s function associated with (-△)m in Rn.Then by using the method of moving planes in integral forms, we prove that there is no positive solution for integral equation (4) in subcritical cases n/n-2m< p< n+2m+a/n-2m.For the non-existence of positive radially symmetrical solutions, we can extend the range to subcritical cases 1<p< n+2m+2a/n-2m.This partially solves an open conjecture posed by Quoc Hung Phan and Philippe Souplet [35].We intend to separate the dissertation into five chapters as following:In chapter one, we introduce the background of higher order elliptical equations and the fundamental methods of the dissertation is also illustrated.In chapter two, we obtain the super poly-harmonic properties of the positive solutions of Hardy-Henon type equation(3), and thus prove Theorem 2.In chapter three, based on Theorem 2, we will establish the equivalence between the integral equations and PDEs and thus prove Theorem 1.In chapter four, we illustrate the Kelvin transform, then we introduce an equiva-lence of the Hardy - Littlewood - Sobolev inequality and Holder inequality, then we use the moving plane method in integral forms to establish a Liouville theorem for Integral equations(4).In chapter five, we will show that in subcritical cases, PDEs(1) have no positive radially symmetric solutions and thus prove Theorem 4.