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.
Categories: math.AP