High regularity of the solution to the singular elliptic p(⋅)-Laplacian system

We study the regularity properties of solutions to the non-homogeneous singular p(x)-Laplacian system in a bounded domain of Rn. Under suitable restrictions on the range of p(x), we construct a W2,r solution, with r>n, that implies the Hölder continuity of the gradient. Moreover, assuming just p(x)∈(1,2) we prove that the second derivatives belong to L2