Variable Lorentz estimate for nonlinear elliptic equations with partially regular nonlinearities

Abstract

We prove global Calder\'on-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-H\"older continuity.