LIOUVILLE TYPE THEOREMS FOR POLY-HARMONIC NAVIER PROBLEMS

In this paper we consider the following semi-linear poly-harmonic equation with Navier boundary conditions on the half space R-n(+): {(-Delta)(alpha/2)u = u(p), in R-+(n) u = -Delta u = ... = (-Delta)(alpha/2-1)u = 0, on partial derivative R-+(n) (1) where alpha is any even number between 0 and n, and p > 1. First we prove that (1) is equivalent to the following integral equation u(x) = integral(R+n) G (x, y, alpha) u(p)(y) dy; x is an element of R-+(n), (2) under some very mild growth condition, where G (x; y; alpha) is the Green's function associated with the same Navier boundary conditions on the half-space. Then by combining the method of moving planes in integral forms with a certain type of Kelvin transform, we obtain the non-existence of positive solutions for integral equation (2) in both subcritical and critical cases under only local integrability conditions. This remarkably weaken the global integrability assumptions on solutions in paper [3]. Our results on integral equation (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 significantly weaken the growth conditions on the solutions.