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

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

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