A non-existence result for higher-order Hardy-Hénon inequality on punctured balls

Abstract

Let  and  be two positive integers such that . Let  and  be real such that  and . In this note, we are mainly concerned with non-negative and classical solutions of the high-order harmonic inequality

on the punctured ball . Using the method of test functions, the Hölder's inequality, and integral estimates, we will prove that this inequality has no  positive solution satisfying some sufficient conditions. It should be mentioned that our result, see Theorem 1.1 in the next section, in the high-order setting is analogous to that of Laptev for the case .