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

Let n and m be two positive integers such that \(n>2m≥ 4\). Let and be real such that \(σ<-2m\) and p > 1 . In this note, we are mainly concerned with non-negative and classical solutions of the high-order harmonic inequality \[(-Δ)ᵐ u ≥ |x|σ uᵖ ,\] on the punctured ball \(B_R\{0\}⊂ R^n\). Using the method of test functions, the Hölder's inequality, and integral estimates, we will prove that this inequality has no \(C²ᵐ\) 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 \(m=2\).