A characterization related toSchrödinger equations on Riemannian manifolds

In this paper, we consider the following problem: {-Delta(g)u +V(x)u =lambda alpha(x)f(u), in M, u >= 0, in M, (p(lambda)) u -> 0, as d(g)(x(0), x) -> infinity, where (M, g) is a N-dimensional (N >= 3), non-compact Riemannian manifold with asymptotically non-negative Ricci curvature, lambda is a real parameter, V is a positive coercive potential, alpha is a bounded function and f is a suitable nonlinearity. By using variational methods, we prove a characterization result for existence of solutions for (P-lambda).