Integrability of solutions for the integral system

Abstract

This paper is concerned with the following integral system involving the Riesz potential and the Wolff potential(1){u(x)=Wβ,γ(up−1v)(x),x∈Rn,v(x)=Iα(up)(x),x∈Rn, where p>1, 1<γ≤2, α∈(0,n) and γβ∈(0,n). By the Hardy-Littlewood-Sobolev inequality and the regularity lifting lemma, we obtain the optimal integrability of integrable solutions (u,v). This integrability is the key ingredient to study the decay rate at infinity of the solutions.