Symmetry and nonexistence of positive solutions to an integral system with weighted functions  

作　　者：DOU JingBo QU ChangZheng HAN YaZhou 

机构地区：[1]Center for Nonlinear Studies,Northwest University,Xi'an 710069,China [2]School of Statistics,Xi'an University of Finance and Economics,Xi'an 710100,China [3]Department of Mathematics,College of Science,China Jiliang University,Hangzhou 310018,China

出　　处：《Science China Mathematics》2011年第4期753-768,共16页中国科学：数学（英文版）

基　　金：supported by Chinese National Science Fund for Distinguished Young Scholars (Grant No.10925104);National Natural Science Foundation of China (Grant No.11001221);the Foundation of Shaanxi Province Education Department (Grant No. 2010JK549);the Foundation of Xi’an Statistical Research Institute (Grant No.10JD04)

摘　　要：Consider the system of integral equations with weighted functions in Rn,{u(x) =∫Rn|x-y|α-nQ(y)v(y)qdy1,v(x)=∫Rn|x-y|α-nK(y)u(y)pdy,where 0 < α < n,1/(p+1) + 1/(q+1)≥(n-α)/n1,α/(n-α) < p1q < ∞1,Q(x) and K(x) satisfy some suitable conditions.It is shown that every positive regular solution(u(x)1,v(x)) is symmetric about some plane by developing the moving plane method in an integral form.Moreover,regularity of the solution is studied.Finally,the nonexistence of positive solutions to the system in the case 0 < p1q <(n+α)/(n-α) is also discussed.Consider the system of integral equations with weighted functions in Rn,{u（x） =∫Rn｜x-y｜α-nQ（y）v（y）qdy1,v（x）=∫Rn｜x-y｜α-nK（y）u（y）pdy,where 0 α n,1/（p＋1） ＋ 1/（q＋1）≥（n-α）/n1,α/（n-α） p1q ∞1,Q（x） and K（x） satisfy some suitable conditions.It is shown that every positive regular solution（u（x）1,v（x）） is symmetric about some plane by developing the moving plane method in an integral form.Moreover,regularity of the solution is studied.Finally,the nonexistence of positive solutions to the system in the case 0 p1q （n＋α）/（n-α） is also discussed.

关 键 词：Hardy-Littlewood-Sobolev inequality system of integral equations SYMMETRY REGULARITY conformally invariant property 

分 类 号：O175.8[理学—数学] TP273[理学—基础数学]