Super poly-harmonic property of solutions for Navier boundary problems on a half space

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