Local behavior of solutions of quasilinear elliptic equations with general structure

Abstract:

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