  In this paper, we are concerned with the critical order H\'{e}non-Lane-Emden
type equations with Navier boundary condition on a half space $\mathbb{R}^n_+$:
\begin{equation}\label{NPDE0}\\\begin{cases} (-\Delta)^{\frac{n}{2}}
u(x)=f(x,u(x)),\ u(x)\geq0,\ x\in\mathbb{R}^{n}_+, \\ u=(-\Delta)u = \cdots =
(-\Delta)^{\frac{n}{2}-1}u = 0,\ x\in\partial\mathbb{R}^{n}_+,
\end{cases}\end{equation} where $u\in C^{n}(\mathbb{R}^{n}_+)\cap
C^{n-2}(\overline{\mathbb{R}^{n}_+})$ and $n\geq2$ is even. We first consider
the typical case $f(x,u)=|x|^{a}u^{p}$ with $0\leq a<\infty$ and $1<p<\infty$.
We prove the super poly-harmonic properties and establish the equivalence
between (0.1) and the corresponding integral equations
\begin{equation}\label{IE0} u(x)=\int_{\mathbb{R}^{n}_+}G(x,y)f(y,u(y))dy,
\end{equation} where $G(x,y)$ denotes the Green's function for
$(-\Delta)^{\frac{n}{2}}$ on $\mathbb{R}^n_+$ with Navier boundary conditions.
Then, we establish Liouville theorem for (0.2) via ``the method of scaling
spheres" developed initially in \cite{DQ0} by Dai and Qin, and hence we obtain
the Liouville theorem for (0.1) on $\mathbb{R}^n_+$. As an application of the
Liouville theorem on $\mathbb{R}^n_+$ (Theorem 1.6) and Liouville theorems in
$\mathbb{R}^{n}$ established in Chen, Dai and Qin [4] for $n\geq4$ and
Bidaut-V\'{e}ron and Giacomini [1] for $n=2$, we derive a priori estimates and
existence of positive solutions to critical order Lane-Emden equations in
bounded domains for all $n\geq2$ and $1<p<\infty$. Extensions to IEs and PDEs
with general nonlinearities $f(x,u)$ are also included.
