CZ estimates for some linear and nonlinear elliptic systems with discontinuous coefficients

The aim of this paper is to derive some Calderón-Zygmund estimates for local solutions u of nonlinear elliptic systems of the type aligned divA(x, Du) = div|G|p - 2G in Ω ⊂ Rⁿ, aligned div A ( x , D u ) = div | G | p - 2 G in Ω ⊂ R n , where Ω ⊂ Rⁿ Ω ⊂ R n is a bounded domain. We assume that u: Ω → RN u : Ω → R N belongs to a weighted Sobolev space Wloc1, p W loc 1 , p , with p ∈ ( 2n/n + 2, 2) p ∈ 2 n n + 2 , 2 , G G belongs to a weighted Llocᵖ L loc p space and x → A(x, ξ ) x → A ( x , ξ ) has growth coefficients in the John and Nirenberg space BMO . As an application of similar techniques, we also consider weak solutions u of linear elliptic systems of the type aligned div(A(x)Du) = div\, G, aligned