Global gradient estimates for elliptic equations of p(x)-Laplacian type with BMO nonlinearity

Abstract

Abstract We consider a nonlinear elliptic problem in divergence form, with nonstandard growth conditions, on a bounded domain. We obtain the global Calderón–Zygmund type gradient estimates for the weak solution of such a problem in the setting of Lebesgue and Sobolev spaces with variable p(x) exponents, in the case that the nonlinearity of the coefficients is allowed to be discontinuous and the domain goes beyond the Lipschitz category. We assume that the nonlinearity has small BMO semi-norms and the boundary of the domain satisfies the so-called δ-Reifenberg flatness condition. These conditions on the nonlinearity and the boundary are weaker than those reported in other studies in the literature.