  We prove the existence of a solution of the Yamabe equation on complete
manifolds with finite volume and positive Yamabe invariant. In order to
circumvent the standard methods on closed manifolds which heavily rely on
global (compact) Sobolev embeddings we approximate the solution by
eigenfunctions of certain conformal complete metrics.
  This also gives rise to a new proof of the well-known result for closed
manifolds and positive Yamabe invariant.
