Liouville type theorems for poly-harmonic Navier problems

In this paper we consider the followingsemi-linear poly-harmonic equation with Navier boundary conditionson the half space $R^n_+$:\begin{equation}\left\{\begin{array}{l}(-\triangle)^{\frac{\alpha}{2}} u=u^p,\ \ \ \ \ \:\:\: \:\:\:\:\:\\:\:\ \ \ \ \ \ \ \ \ \ \ \ \:\:\:\:\ \mbox{in}\,\ R^n_+,\\ u=-\triangle u=\cdots=(-\triangle)^{\frac{\alpha}{2}-1}u=0, \ \ \ \mbox{on}\ \partial R^n_+, \end{array} \right. \label{phe1} \end{equation}where $\alpha$ is any even number between $0$ and $n$, and $p>1$.   First we prove that (1) is equivalent to the followingintegral equation\begin{equation}u(x)=\int_{R^n_+}G(x,y,\alpha) u^p(y)dy,\,\,\,\,\, x\in\,R^n_+,\label{ie0} \end{equation}under some very mild growth condition, where $G(x, y,\alpha)$ is the Green's function associated with thesame Navier boundary conditions on the half-space .   Then by combining the method of moving planes in integral formswith a certain type of Kelvin transform, we obtain the non-existenceof positive solutions for integral equation (2) in bothsubcritical and critical cases under only local integrabilityconditions. This remarkably weaken the global integrabilityassumptions on solutions in paper [3]. Our results on integralequation (2) are valid for all real values $\alpha$ between$0$ and $n$.   Finally, we establish a Liouville type theorem for PDE (1),and this generalizes Guo and Liu's result [21] by significantlyweaken the growth conditions on the solutions.