On Green functions of second-order elliptic operators on Riemannian manifolds: The critical case

TL;DR

This paper investigates the existence and uniqueness of Green functions for critical second-order elliptic operators on noncompact Riemannian manifolds, extending classical results and analyzing their behavior at infinity.

Contribution

It establishes the existence of Green functions dominated by ground states for critical operators and characterizes their uniqueness and asymptotic behavior.

Findings

Existence of Green functions dominated by ground states for critical operators.

Uniqueness of Green functions up to a product of ground states.

Description of Green function behavior at infinity under certain conditions.

Abstract

Let P be a second-order, linear, elliptic operator with real coefficients which is defined on a noncompact and connected Riemannian manifold M. It is well known that the equation Pu = 0 in M admits a positive supersolution which is not a solution if and only if P admits a unique positive minimal Green function on M, and in this case, P is said to be subcritical in M. If P does not admit a positive Green function but admits a global positive solution, then such a solution is called a ground state of P in M, and P is said to be critical in M. We prove for a critical operator P in M, the existence of a Green function which is dominated above by the ground state of P away from the singularity. Moreover, in a certain class of Green functions, such a Green function is unique, up to an addition of a product of the ground states of P and P^{\star}. Under some further assumptions, we describe the behaviour at infinity of such a Green function. This result extends and sharpens the celebrated result of P. Li and L.-F. Tam concerning the existence of a symmetric Green function for the Laplace-Beltrami operator on a smooth and complete Riemannian manifold M.