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

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