On the solvability of boundary value problems for semilinear elliptic equations on noncompact riemannian manifold

In this paper we study the solvability of certain boundary and external boundary value problems for semilinear elliptic equations (5) ??? = ??(|??|)??, where ??(??) - nonnegative, nondecreasing continuously differentiable function for ?? ? 0 on arbitrary non-compact Riemannian manifolds. In this article we compare the behavior of nonbounded functions "at infinity". In our research we use a new approach wich is based on the consideration of equivalence classes of functions on ?? (this approach for bounded solutions has been realized in [8]). Let ?? be an arbitrary smooth connected noncompact Riemannian manifold without boundary and let {????}??? =1 be an exhaustion of ??, i.e., a sequence of precompact open subsets of ?? such that ???? ? ????+1 and ?? = ????? =1 ????. Throughout the sequel, we assume that boundaries ?????? are ??1-smooth submanifolds. Let ??1 and ??2 be arbitrary continuous functions on ??. Say that ??1 and ??2 are equivalent on ?? and write ??1 ? ??2 if for some exhaustion {????}??? =1 of ?? we have lim ??>? sup ??????? |??1 ? ??2| = 0. It is easy to verify that the relation " ? " is an equivalence which does not depend on the choice of the exhaustion of the manifold and so partitions the set of all continuous functions on ?? into equivalence classes. Denote the equivalence class of a function ?? by [??]. Let ?? ? ?? be an arbitrary connected compact subset and the boundary of ?? is a ??1-smooth submanifold. Assume that the interior of ?? is non-empty and ?? ? ???? for all ??. Observe that if the manifold ?? has compact boundary or there is a natural geometric compactification of ?? (for example, on manifolds of negative sectional curvature or spherically symmetric manifolds) which adds the boundary at infinity, then this approach leads naturally to the classical statement of the Dirichlet problem. Denote by ???? the solution of equation (5) in ???? ? ?? which satisfies to conditions ????|???? = 1, ????|?????? = 0. Using the maximum principle, we can easily verify that the sequence ???? is uniformly bounded on ????? and so is compact in the class of twice continuously differentiable functions over every compact subset ?? ? ?? ? ??. Moreover, as ?? > ? this sequence increases monotonically and converges on ?? ? ?? to a solution of equation (5) ?? = lim ??>? ????, 0 0 such that 0 ? ??(??) ? ?? with ?? ? 0. We put in the equation (2) ??(??) ? ??. Also let ?? is an asymptotically nonnegative function on ??. We now formulate the main result. Theorem 1. Let ?? be an ?-strict manifold. Suppose that for every positive constant the exterior boundary value problems for the equations (1) and (2) are solvable on ?? ? ?? with boundary conditions of class [??]. Then: 1. for every continuous function ?(??) ? 0 on ???? the exterior boundary value problem for (5) is solvable on ?? ? ?? with boundary conditions of class [??]; 2. the boundary value problem for (5) is solvable on ?? with boundary conditions of class [??].