Gradient estimates for anisotropic elliptic equations

We study a class of elliptic equations with different degeneracies in different directions such as |ux| m-2 ux x + |uy| q-2 uy 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.