  In this paper, we are concerned with the non-critical higher order
Lane-Emden-Hardy equations \begin{equation*}
  (-\Delta)^{m}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in}
\,\,\, \mathbb{R}^{n} \end{equation*} with $n\geq3$, $1\leq m<\frac{n}{2}$,
$0\leq a<2m$, $1<p<\frac{n+2m-2a}{n-2m}$ if $0\leq a<2$, and $1<p<\infty$ if
$2\leq a<2m$. We prove Liouville theorems for nonnegative classical solutions
to the above Lane-Emden-Hardy equations (Theorem \ref{Thm0}), that is, the
unique nonnegative solution is $u\equiv0$. As an application, we derive a
priori estimates and existence of positive solutions to non-critical higher
order Lane-Emden equations in bounded domains (Theorem \ref{Thm1} and
\ref{Thm2}). The results for critical order Hardy-H\'{e}non equations have been
established by Chen, Dai and Qin \cite{CDQ} recently.
