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 Rⁿ₊:{equation}\{{array}{l}(-)α/2 u=u^p,\ \ \ \ \ \:\:\: \:\:\:\:\:\\:\:\ \ \ \ \ \ \ \ \ \ \ \ \:\:\:\:\ in\,\ R^n_+,\\ u=- u=⋯=(-)α/2-1u=0, \ \ \ on\ ∂ R^n_+, {array} . {phe1} {equation}where α is any even number between $0$ and n, and $p>1$.   First we prove that (1) is equivalent to the followingintegral equation{equation}u(x)=∫R^n_+G(x,y,α) u^p(y)dy,\,\,\,\,\, x∈\,R^n_+,{ie0} {equation}under some very mild growth condition, where G(x, y,α) 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 α 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.