On solvability of boundary value problems of the Poisson equation on non-compact Riemannian manifolds

This article is devoted to the investigation of the behavior of solutions of the Poisson equation in relation to the geometry of the manifold in question. Such problems originate in the classification theory of non-compact Riemannian surfaces and manifolds. For a noncompact Riemann surface, the well-known problem of conformal type identification can be stated as follows: Does a nontrivial positive superharmonic function exist on this surface? Many questions of this kind fit into the pattern of a Liouville-type theorem saying that the space of bounded solutions of some elliptic equation is trivial. However, the class of manifolds admitting nontrivial solutions of some elliptic equations is wide. For example, conditions ensuring the solvability of the Dirichlet problem with continuous boundary conditions “at infinity” for several noncompact manifolds has been found in many papers (see, e.g., [12; 18; 21]). Notice that the very statement of the Dirichlet problem on such manifolds could turn out nontrivial, since it is unclear how we should interpret the boundary data. In this article we study questions of existence and belonging to given functional class of bounded solutions of the Poisson equation Δ𝑢 = 𝑔(𝑥), (1) where 𝑔(𝑥) ∈ (Ω) for any subset Ω ⊂⊂ 𝑀, 0