Gradient estimates for anisotropic elliptic equations

Abstract

We study a class of elliptic equations with different degeneracies in different directions such as $$ \left( |u_x|^{m-2}u_x\right)_x + \left( |u_y|^{q-2}u_y\right)_y =0 $$ with unequal parameters $m$ and $q$, both greater than one. We show that any bounded solution of such an equation must have a bounded gradient. The ideas also apply to a much more general class of degenerate equations, which we describe in some detail.