LAUSR

Abstract We prove a Harnack inequality for nonnegative strong solutions to degenerate and singular elliptic PDEs modeled after certain convex functions and in the presence of unbounded drifts. Our main theorem extends the Harnack inequality for the linearized Monge–Ampere equation due to Caffarelli and Gutierrez and it is related, although under different hypotheses, to a recent work by N.Q. Le. Since our results are shown to apply to the convex functions | x | p with p ≥ 2 and their tensor sums, the degenerate elliptic operators that we can consider include subelliptic Grushin and Grushin-like operators as well as a recent example by A. Montanari of a nondivergence-form subelliptic operator arising from the geometric theory of several complex variables. In the light of these applications, it follows that the Monge–Ampere quasi-metric structure can be regarded as an alternative to the usual Carnot–Caratheodory metric in the study of certain subelliptic PDEs.