Gradient Estimates for Singular-Degenerate Elliptic Equations with Solution-Dependent Coefficients

We study a conormal boundary value problem for a class of singular-degenerate quasilinear elliptic equations in divergence form in bounded domains. The leading terms are measurable in the spatial variable and depend nonlinearly on both the solution and its gradient. The singularity or degeneracy is governed by a power of the distance to the domain boundary, with exponent greater than negative one. In particular, the corresponding weights need not belong to the Muckenhoupt $A_2$ class. Assuming that the weighted mean oscillations of the leading terms in the spatial variable are sufficiently small on small balls and that the domains have Lipschitz boundaries with sufficiently small Lipschitz constants, we establish global weighted $W^{1,p}$-estimates for weak solutions with bounded weighted mean oscillation. Our results extend recent estimates for equations whose leading terms are independent of the solution itself.