一个带不确定权的积分方程组解的对称性

A symmetry of solutions for a system of integral equations with the indefinite weight functions

结合积分形式移动平面法的思想,讨论Rn上积分方程组u(x)=∫Rn|x-y|α-na(y)v(y)qdy,v(x)=∫Rn|x-y|α-nb(y)u(y)pdy的正解关于某一点的对称性和单调性,其中0<α<n,p,q>1,p+11+q+11=n n-α,a(x)和b(x)满足一些对称性、单调性.

This paper is devoted to the study of the positive solutions of the following system of integral equations in Rn： u（x）=∫Rn｜x-y｜α-na（y）v（y）qdy,v（x）=∫Rn｜x-y｜α-nb（y）u（y）pdy,where 0αn,p,q1,1p＋1＋1q＋1=n-αn,a（x） and b（x） satisfy some symmetric and integral conditions.Inspired by the idea of the method of moving planes in integral forms,it has been proved that all the solutions are symmetric and monotone decreasing about some point.