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

In this paper we give a new and direct proof of local Lq estimates for the nonhomogeneous p(x) -Laplacian elliptic equation under some proper conditions on p(x) > 1 . We prove that |f|p(x) ∈ Lqloc =⇒ |∇u|p(x) ∈ Lqloc for any q 1 for weak solutions of div ( |∇u|p(x)−2∇u ) = div ( |f|p(x)−2f ) in Ω. Mathematics subject classification (2010): 35J60, 35J70.