Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R~n,u(x)=-?u(x)=…=(-?)u(x)=0,on ?R~n,(0.1)where m is any positive integer satisfying 0<2m<n.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu>0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c∫R~n(1/|x-y|-1/|x~*-y|)u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x,-x) is the reflection of the point x about the plane R.Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).