Local Behavior of Solutions of Quasilinear Elliptic Equations With General Structure

This paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form <!-- MATH $\operatorname{div} A(x,u,\nabla u) + B(x,u,\nabla u)$ --> , where and are Borel measurable, are solutions to the equation <!-- MATH $\operatorname{div} A(x,u,\nabla u) + B(x,u,\nabla u) = \mu$ --> for some nonnegative Radon measure . Among other things, it is shown that if is a Hölder continuous solution to this equation, then the measure satisfies the growth property <!-- MATH $\mu [B(x,r)] \leqslant M{r^{n - p + \varepsilon }}$ --> for all balls in <!-- MATH ${{\mathbf{R}}^n}$ --> . Here <!-- MATH $\varepsilon$ --> depends on the Hölder exponent of while 1$"> is given by the structure of the differential operator. Conversely, if is assumed to satisfy this growth condition, then it is shown that satisfies a Harnack-type inequality, thus proving that is locally bounded. Under the additional assumption that is strongly monotonic, it is shown that is Hölder continuous.