Variable Lorentz estimate for nonlinear elliptic equations with partially regular nonlinearities

We prove global Calderón-Zygmund type estimate in Lorentz spaces for variable power of the gradients to weak solution of nonlinear elliptic equations in a non-smooth domain. We mainly assume that the nonlinearities are merely measurable in one of the spatial variables and have sufficiently small BMO semi-norm in the other variables, the boundary of domain belongs to Reifenberg flatness, and the variable exponents p(x) satisfy log-Hölder continuity.