Weighted gradient estimates for the class of very singular p-Laplace system

Abstract Let n ∈ { 2 , 3 , 4 , … } , N ∈ { 1 , 2 , 3 , … } and p ∈ ( 1 , 2 − 1 n ] . Let β ∈ ( 1 , ∞ ) be such that n p n − p β ′ n n ( 2 − p ) − 1 and f ∈ L β ( R n , R N ) . Consider the p-Laplace system − Δ p u = − div ( | D u | p − 2 D u ) = f in R n . We obtain a weighted gradient estimate for weak solutions of this system.