Harnack inequality for quasilinear elliptic equations on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equationsdiv(F′(|∇u|)|∇u|∇u)=h of p-Laplacian type on Riemannian manifolds, where an even function F∈C1(R)∩C2(0,∞) is supposed to be strictly convex on (0,∞). Under the assumption that either F∈C2(R) or its convex conjugate F⁎∈C2(R) with some structural condition, we establish a (locally) uniform ABP type estimate and the Krylov-Safonov type Harnack inequality on Riemannian manifolds with the use of an intrinsic geometric quantity to the operator. Here, the C2-regularities of F and F⁎ account for degenerate and singular operators, respectively.