Regularity of solutions for an integral system of Wolff type

We consider the fully nonlinear integral systems involving Wolff potentials:equation(1){u(x)=Wβ,γ(vq)(x),x∈Rn,v(x)=Wβ,γ(up)(x),x∈Rn; whereWβ,γ(f)(x)=∫0∞[∫Bt(x)f(y)dytn−βγ]1γ−1dtt.This system includes many known systems as special cases, in particular, when β=α2 and γ=2γ=2, system (1) reduces toequation(2){u(x)=∫Rn1|x−y|n−αv(y)qdy,x∈Rn,v(x)=∫Rn1|x−y|n−αu(y)pdy,x∈Rn. The solutions (u,v)(u,v) of (2) are critical points of the functional associated with the well-known Hardy–Littlewood–Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs{(−Δ)α/2u=vq,in Rn,(−Δ)α/2v=up,in Rn, which comprises the well-known Lane–Emden system and Yamabe equation.We obtain integrability and regularity for the positive solutions to systems (1). A regularity lifting method by contracting operators is used in proving the integrability, and while deriving the Lipschitz continuity, a brand new idea – Lifting Regularity by Shrinking Operators is introduced. We hope to see many more applications of this new idea in lifting regularities of solutions for nonlinear problems.