Weighted gradient estimates to nonlinear elliptic equations of p(x)-growth with measure data

Abstract

We consider a nonlinear elliptic equation of the form $-\text{div}A(x,u,Du)=\mu$, where the principle part depends on the solution itself and the right-hand data μ is a signed Radon measure. The associated nonlinearity is assumed to satisfy the $(\delta ,R_0)$-BMO condition in x and the Lipschitz continuity condition in u, and its growth in Du is like the $p\left(x\right)$-Laplacian, while the boundary of underlying domain is assumed to be Reifenberg flat. We establish an optimal global Calderón-Zygmund type estimate in weighted Lorentz spaces for the gradients of very weak solutions to such a measure data problem. This is achieved by developing the perturbation method and modifying the weighted Vitali type covering argument.