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