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.

中文翻译：

---

关于黎曼流形上二阶椭圆算子的格林函数：临界情况

令 P 是一个二阶、线性、椭圆算子，其实系数定义在一个非紧且连通的黎曼流形 M 上。众所周知，M 中的方程 Pu = 0 承认一个正超解，如果和仅当 P 承认 M 上唯一的正最小格林函数，并且在这种情况下，称 P 在 M 中是次临界的。如果 P 不承认正格林函数但承认全局正解，则这种解称为 a M 中 P 的基态，并且 P 被称为 M 中的临界状态。我们证明了 M 中的临界算子 P，存在一个格林函数，该函数由远离奇点的 P 的基态支配。此外，在某一类格林函数中，这样的格林函数是唯一的，最多是 P 和 P^{\star} 的基态的乘积相加。在一些进一步的假设下，我们描述了这种格林函数在无穷远处的行为。这一结果扩展并突出了 P. Li 和 L.-F. 的著名结果。Tam 关于在光滑且完备的黎曼流形 M 上 Laplace-Beltrami 算子的对称格林函数的存在性。