In this paper, we are concerned with the critical (i.e., n-th) order Hénon-Lane-Emden type equations with Navier boundary conditions on a half space ${\mathbb{R}}_{+}^n$:(0.1)$\{\begin{array}{c}{(-\mathrm{\Delta })}^{\frac{n}{2}}u(x)=f(x,u(x)),u(x)\ge 0,x\in {\mathbb{R}}_{+}^n,\hfill \\ u(x)=-\mathrm{\Delta }u(x)=\cdots ={(-\mathrm{\Delta })}^{\frac{n}{2}-1}u(x)=0,x\in \partial {\mathbb{R}}_{+}^n,\hfill \end{array}$ where $u\in C^n({\mathbb{R}}_{+}^n)\cap C^{n-2}(\overline{{\mathbb{R}}_{+}^n})$ and $n\ge 2$ is even. We first consider the typical case $f(x,u)=|x{|}^a{u}^p$ with $0\le a<\infty$ and $1<p<\infty$. We prove the super poly-harmonic properties and establish the equivalence between (0.1) and the corresponding integral equations(0.2)$u(x)=\underset{{\mathbb{R}}_{+}^n}{\int }{G}^{+}(x,y)f(y,u(y))dy,$ where $G^{+}(x,y)$ denotes the Green function for ${(-\mathrm{\Delta })}^{\frac{n}{2}}$ on ${\mathbb{R}}_{+}^n$ with Navier boundary conditions. Then, we establish Liouville theorem for (0.2) and hence 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 ${\mathbb{R}}^n$, we derive a priori estimates via blowing-up methods for solutions (possibly change signs) to Navier problems involving critical order uniformly elliptic operators L. Consequently, by using the Leray-Schauder fixed point theorem, we derive existence of positive solutions to critical order Lane-Emden equations in bounded domains for all $n\ge 2$ and $1<p<\infty$. In contrast to the subcritical order cases, our results seem to be the first work on Navier problems for critical order equations on ${\mathbb{R}}_{+}^n$, which is the critical-order counterpart to those results on subcritical order cases in [6], [20], [21]. Extensions to IEs and PDEs with general nonlinearities $f(x,u)$ are also included. Surprisingly, there are no growth conditions on u and hence $f(x,u)$ can grow exponentially (or even faster) on u.

在本文中，我们关注在半空间上具有 Navier 边界条件的临界（即 n 阶） Hénon-Lane-Emden 型方程${\mathbb{电阻}}_{+}^n$: (0.1)$\{\begin{array}{c}{(-\mathrm{\Delta })}^{\frac{n}{2}}你(X)=F(X,你(X)),你(X)\ge 0,X\in {\mathbb{电阻}}_{+}^n,\hfill \\ 你(X)=-\mathrm{\Delta }你(X)=\cdots ={(-\mathrm{\Delta })}^{\frac{n}{2}-1}你(X)=0,X\in \partial {\mathbb{电阻}}_{+}^n,\hfill \end{array}$ 在哪里 $你\in C^n({\mathbb{电阻}}_{+}^n)\cap C^{n-2}(\overline{{\mathbb{电阻}}_{+}^n})$ 和 $n\ge 2$甚至。我们首先考虑典型案例$F(X,你)=|X{|}^{\mathrm{一种}}{你}^{磷}$ 和 $0\le \mathrm{一种}<\infty$ 和 $1<磷<\infty$. 我们证明了超多谐波性质并建立了（0.1）和相应的积分方程（0.2）之间的等价性$你(X)=\underset{{\mathbb{电阻}}_{+}^n}{\int }{G}^{+}(X,是)F(是,你(是))d是,$ 在哪里 $G^{+}(X,是)$ 表示格林函数 ${(-\mathrm{\Delta })}^{\frac{n}{2}}$ 在 ${\mathbb{电阻}}_{+}^n$纳维边界条件。然后，我们建立刘维定理为（0.2），并因此获得刘维定理（0.1）上${\mathbb{电阻}}_{+}^n$. 作为刘维尔定理的应用${\mathbb{电阻}}_{+}^n$ （定理 1.6）和 Liouville 定理 ${\mathbb{电阻}}^n$，我们通过爆炸方法（可能是改变符号）来推导出先验估计，以解决涉及临界阶一致椭圆算子 L 的Navier 问题。因此，通过使用Leray-Schauder 不动点定理，我们推导出所有有界域中临界阶 Lane-Emden 方程的正解的存在性$n\ge 2$ 和 $1<磷<\infty$. 与亚临界阶情况相反，我们的结果似乎是关于临界阶方程的纳维问题的第一项工作${\mathbb{电阻}}_{+}^n$，这是[6]、[20]、[21]中亚临界顺序情况下的那些结果的临界顺序对应物。具有一般非线性的 IE 和 PDE 的扩展$F(X,你)$也包括在内。令人惊讶的是，u上没有生长条件，因此$F(X,你)$可以在u上呈指数增长（甚至更快）。