Regularity of solutions for an integral system of Wolff type

Abstract

We consider the fully nonlinear integral systems involving Wolff potentials: (1) { u ( x ) = W β , γ ( v q ) ( x ) , x ∈ R n , v ( x ) = W β , γ ( u p ) ( x ) , x ∈ R n ; where W β , γ ( f ) ( x ) = ∫ 0 ∞ [ ∫ B t ( x ) f ( y ) d y t n − β γ ] 1 γ − 1 d t t . This system includes many known systems as special cases, in particular, when β = α 2 and γ = 2 , system (1) reduces to (2) { u ( x ) = ∫ R n 1 | x − y | n − α v ( y ) q d y , x ∈ R n , v ( x ) = ∫ R n 1 | x − y | n − α u ( y ) p d y , x ∈ R n . The solutions ( 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 / 2 u = v q , in R n , ( − Δ ) α / 2 v = u p , in R n , 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.