  In this paper, we are devoted to studying the positive solutions of the
following higher order Hardy-H\'enon equation $$
(-\Delta)^{m}u=|x|^{\alpha}u^{p} \quad\mbox{in}~
B_{1}\setminus\{0\}\subset\mathbb{R}^{n} $$ with an isolated singularity at the
origin, where $\alpha>-2m$, $m\geq1$ is an integer and $n>2m$. For
$1<p<\frac{n+2m}{n-2m}$, singularity and decay estimates of solutions will be
given. For $\frac{n+\alpha}{n-2m}<p<\frac{n+2m}{n-2m}$ with $-2m<\alpha<2m$, we
show the super polyharmonic properties of solutions near the singularity, which
are essential tools in the study of polyharmonic equation. Using these
properties, a classification of isolated singularities of positive solutions is
established for the fourth order case, i.e., $m=2$. Moreover, when $m=2$,
$\frac{n+\alpha}{n-4}<p<\frac{n+4+\alpha}{n-4}$ and $p\neq
\frac{n+4+2\alpha}{n-4}$ with $-4<\alpha\leq0$, we obtain the precise behavior
of solutions near the singularity, i.e., either $x=0$ is a removable
singularity or
$$\lim_{|x|\rightarrow0}|x|^{\frac{4+\alpha}{p-1}}u(x)=[A_{0}]^{\frac{1}{p-1}},$$
where $A_0>0$ is an exact constant.

Analysis of PDEs