Existence and non-existence results for the higher order Hardy–Hénon equations revisited

This paper is devoted to studies of non-negative, non-trivial (classical, punctured, or distributional) solutions the higher order Hardy-H\'enon equations \[ (-\Delta)^m u = |x|^\sigma u^p \] in $\mathbf R^n$ with $p > 1$. We show that condition n - 2m \frac{2m+\sigma}{p-1} >0 necessary for existence distributional solutions. For $n \geq 2m$ and $\sigma -2m$, we prove any solution satisfies an integral equation a weak super polyharmonic property. establish some sufficient conditions punctured classical be solution. As application, if there no non-trivial, 1 < p \frac{n+2m+2\sigma}{n-2m}. At last, 2m$, -2m$ $$p \frac{n+2m+2\sigma}{n-2m},$$ exist positive, radially symmetric, equation.