Some symmetry results for integral equations involving Wolff potential on bounded domains

We investigate systems of integral equations involving Wolff potential on a bounded domain such as {u(x)=∫0∞[∫Bt(x)∩Ωua(y)vb(y)dytn−αγ]1γ−1dtt,x∈Ω,v(x)=∫0∞[∫Bt(x)∩Ωuc(y)vd(y)dytn−βκ]1κ−1dtt,x∈Ω where α,β,γ,κ,a,b,c,dα,β,γ,κ,a,b,c,d are constants. We show that uu and vv are constants on ∂Ω∂Ω if and only if ΩΩ is a ball. This result also provides a characterization for integral equations with Wolff potential.