Weighted gradient estimates for the general elliptic p(x)p(x)-Laplacian equations

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