Optimal Regularity for Elliptic Equations With Measurable Nonlinearities Under Nonstandard Growth

We are concerned with weak solutions of elliptic equations involving measurable nonlinearities with Orlicz growth to address what would be the weakest regularity condition on the associated nonlinearity for the Calderón–Zygmund theory. We prove that the gradient of weak solution is as integrable as the nonhomogeneous term under the assumption that the nonlinearity is only measurable in one of the variables while it has a small BMO assumption in the other variables. To this end, we develop a nonlinear Moser-type iteration argument for such a homogeneous reference problem with one variable–dependent nonlinearity under Orlicz growth to establish $W^{1,q}$–regularity for every $q>1$. Our results open a new path into the comprehensive understanding of the problem with nonstandard growth in the literature of optimal regularity theory in highly nonlinear elliptic and parabolic equations.