On solvability of the boundary value problems for harmonic function on noncompact Riemannian manifolds

We study questions of existence and belonging to the given functional class of solutions of the Laplace-Beltrami equations on a noncompact Riemannian manifold $M$ with no boundary. In the present work we suggest the concept of $\phi$-equivalency in the class of continuous functions and establish some interrelation between problems of existence of solutions of the Laplace-Beltrami equations on $M$ and off some compact $B \subset M$ with the same growth "at infinity". A new conception of $\phi$-equivalence classes of functions on $M$ develops and generalizes the concept of equivalence of function on $M$ and allows us to more accurately estimate the rate of convergence of the solution to boundary conditions.