Bôcher-type theorems for the Poisson’s equation on manifolds with conical metrics

Abstract

We prove a generalized Bocher-type theorem for the nonhomogeneous Laplace’s equations on singular manifolds with conical metrics. More specifically, we give a sharp characterization of the behavior at isolated singularities of a solution bounded on one side for the equation $$\Delta _g u =f$$ ( $$f \in L_g^q$$ with $$q > \frac{n}{2}$$ ). The main results in this paper imply that a nonnegative solution with conical singularities has certain of the attributes of fundamental solutions.