Calderon-Zygmund estimate for asymptotically regular elliptic equations withL(p(x))-logarithmic growth

We study asymptotically regular elliptic equations with-logarithmic growth under the assumptions that the variable exponent satisfies strong log-Holder continuity and the underlying domain is Reifenberg flat. To ensure a global Calderon-Zygmund estimate for such asymptotically regular problem, we make use of approximating the solutions for asymptotically regular problem by the solutions for regular problem while the gradients of the solutions close to infinity.