  The goal of this article is to investigate complete noncompact warped product
gradient Ricci solitons. Nonexistence results, estimates for the warping
function and for its gradient are proven. When the soliton is steady or
expanding these nonexistence results generalize to a broader context certain
pde estimates and rigidity obtained when studying warped product Einstein
manifolds. When the soliton is shrinking, it is presented a nonexistence
theorem with no counterpart in the Einstein case, which is proved using
properties of the first eigenvalue of a weighted Laplacian.
