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Symmetry and regularity of solutions to a system with three-component integral equations

Symmetry and regularity of solutions to a system with three-component integral equations
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摘要 Consider the system with three-component integral equations{u(x) :fRn│x - y│α-nw(y)^v(y)^qdy,v(x) =fRn│x-y│^α-nu(y)^pw(y)^rdy, w(x) =fRn│x - y│α-nv(y)qu(y)Pdy,where 0 〈 a 〈 n, n is a positive constant, p, q and r satisfy some suitable conditions. It is shown that every positive regular solution (u(x), v(x), w(x)) is radially symmetric and monotonic about some point by developing the moving plane method in an integral form. In addition, the regularity of the solutions is also proved by the contraction mapping principle. The conformal invariant property of the system is also investigated. Consider the system with three-component integral equations u(x) = Rn |x y|α nw(y)rv(y)q dy,v(x) = Rn |x y|α nu(y)pw(y)rdy,w(x) = Rn |x y|α nv(y)q u(y)pdy,where 0 < α < n,n is a positive constant,p,q and r satisfy some suitable conditions.It is shown that every positive regular solution(u(x),v(x),w(x)) is radially symmetric and monotonic about some point by developing the moving plane method in an integral form.In addition,the regularity of the solutions is also proved by the contraction mapping principle.The conformal invariant property of the system is also investigated.
出处 《Science China Mathematics》 SCIE 2012年第10期1991-2004,共14页 中国科学:数学(英文版)
基金 supported by National National Science Foundation of China for Distinguished Young Scholars (Grant No. 10925104) National National Science Foundation of China (Grant No.11101319) the Foundation of Shaanxi Province Education Department (Grant No. 2010JK549)
关键词 system of integral equations SYMMETRY REGULARITY conformal invariance 积分方程 三分量 系统 对称性 压缩映射原理 FRN 正规溶液 径向对称
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