LAUSR

Abstract We study viscosity solutions to degenerate and singular elliptic equations div ( F ′ ( | ∇ u | ) | ∇ u | ∇ u ) = h of p-Laplacian type on Riemannian manifolds, where an even function F ∈ C 1 ( R ) ∩ C 2 ( 0 , ∞ ) is supposed to be strictly convex on ( 0 , ∞ ) . Under the assumption that either F ∈ C 2 ( R ) or its convex conjugate F ⁎ ∈ C 2 ( 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 C 2 -regularities of F and F ⁎ account for degenerate and singular operators, respectively.