Weighted polyharmonic equation with Navier boundary conditions in a half space  

作　　者：ZHUO Ran 

机构地区：[1]Department of Mathematical Sciences, Huanghuai University, Zhumadian 463000, China [2]Department of Mathematical Sciences, Yeshiva University,New York, NY 10033, USA

出　　处：《Science China Mathematics》2017年第3期491-510,共20页中国科学：数学（英文版）

摘　　要：We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space：where rn is any positive integer satisfying 0 〈 2m 〈 n. We first prove that the positive solutions of （0.1） are super polyharmonic, i.e.,where x＊ = （x1,... ,Xn-1, --Xn） is the reflection of the point x about the plane Rn-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 which α can 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）.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).

关 键 词：Navier boundary conditions half space super polyharmonic EQUIVALENCE integral equation rotational symmetry NON-EXISTENCE 

分 类 号：O175[理学—数学] O177[理学—基础数学]