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