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)=…=(-?)^(m-1)u(x)=0,on ?...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)=…=(-?)^(m-1)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_n∫R_+~n(1/|x-y|^(n-α)-1/|x~*-y|^(n-α))u^p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R^(n-1).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).展开更多
文摘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)=…=(-?)^(m-1)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_n∫R_+~n(1/|x-y|^(n-α)-1/|x~*-y|^(n-α))u^p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R^(n-1).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).
