In this paper, we consider the following poly-harmonic semi-linear equation with Navier boundary conditions on a half space R+n:(1){(−Δ)mu=up,p>1,m⩾1,u>0,inR+n,u=Δu=⋯=Δm−1u=0,on∂R+n. We first prove that the positive solutions of (1) are super poly-harmonic, i.e.(2)(−Δ)iu>0,i=0,1,…,m−1. Then, based on (2), we establish the equivalence between PDE (1) and the integral equation(3)u(x)=cn∫R+n(1|x−y|n−2m−1|x¯−y|n−2m)up(y)dy, where 1<p<∞ and x¯=(x1,…,−xn) is the reflection of x about the boundary. Combining our equivalence result with previous Liouville type theorems on integral equation (3), we derive the non-existence of positive solutions for problem (1). This in turn enables us to obtain a-priori estimates for the solutions of a family of higher order equations with Navier boundary data on either bounded domains in Rn or on Riemannian manifolds with boundaries. A similar super poly-harmonic results like (6) in the whole space Rn has been obtained by Wei and Xu (1999) [48], however, the method used there can no longer be applied to our situation, hence we introduce some new ideas.