  In this paper we investigate the existence of a solution to the Poisson
equation on complete manifolds with positive spectrum and Ricci curvature
bounded from below. We show that if a function $f$ has decay
$f=O(r^{-1-\varepsilon}) $ for some $\varepsilon >0,$ where $r$ is the distance
function to a fixed point, then the Poisson equation $\Delta u=f$ has a
solution $u$ with at most exponential growth.
  We apply this result on the Poisson equation to study the existence of
harmonic maps between complete manifolds and also existence of Hermitian
Einstein metrics on holomorphic vector bundles over complete manifolds, thus
extending some results of Li-Tam and Ni.
  Assuming that the manifold is simply connected and of Ricci curvature between
two negative constants, we can prove that in fact the Poisson equation has a
bounded solution and we apply this result to the Ricci flow on complete
surfaces.
