Integrability and Symmetry of Positive Integrable Solutions for Weighted Wolff-Type Integral Systems

In this paper, we are concerned with the following weighted integral system involving Wolff potential: { u(x)=R-1(x)W beta,gamma(vq|/y|sigma)(x),u(x)>0,x is an element of R-N , v(x)=R-2(x)W beta,gamma(up/|y|sigma)(x), v(x)>0,x is an element of R-N where gamma>2,beta>0, 0<sigma gamma-1, with gamma-1p+gamma-1,gamma-1q+gamma-1<N-beta gamma N,gamma-1p+gamma-1+gamma-1q+gamma-1=N-beta gamma+sigma N,R1,R2are double bounded in R(N )and W-beta,W-gamma(h)(x):=integral(infinity)(0)[integral Bt(x)h(y)dytN-beta gamma](1/gamma-1) dt/t Firstly, by applying Minkowski's inequality and the regularity lifting lemma, we prove the optimal integrability, boundedness and vanishing property at infinity of integrable solutions for the system. Secondly, we use the method of moving planes in integral forms to prove the radial symmetry of the integrable solutions when R1 equivalent to R2 equivalent to 1in R-N