Classification and symmetry of positive solutions for weighted integral system

Abstract

In this paper, we consider the following weighted integral system u(x)=∫R+n+1xn+1αyn+1βf(v(y))|x−y|λdy,y∈R+n+1,v(y)=∫R+n+1xn+1αyn+1βg(u(x))|x−y|λdx,x∈R+n+1.Without the whole integrability assumptions on u and v, we obtain the classification and symmetry of positive solutions by using a slight variant of the method of moving spheres. Our results improve existing ones in Dou et al. (2017) and Gluck (2020) by introducing several new ideas. Furthermore, we prove the classification and symmetry result for integral system with general weights on whole space Rn+m.