Weighted gradient estimates for the general elliptic -Laplacian equations

In this paper we obtain the weighted L q estimates of the gradients of weak solutions for the general elliptic p ( x ) -Laplacian equations div ( a ( x , ∇ u ) ) = div ( | f | p ( x ) − 2 f ) , under some proper assumptions on a, p, f. We prove that | f | p ( x ) ∈ L w , loc q ⟹ | ∇ u | p ( x ) ∈ L w , loc q for any q > 1 .