  The special nature of gradient Yamabe soliton equation which was first
observed by Cao-Sun-Zhang\cite{CaoSunZhang} shows that a complete gradient
Yamabe soliton with non-constant potential function is either defined on the
Euclidean space with rotational symmetry, or on the warped product of the real
line with a manifold of constant scalar curvature. In this paper we consider
the classification in the latter case. We show that a complete gradient steady
Yamabe soliton on warped product is necessarily isometric to the Riemannian
product. In the shrinking case, we show that there is a continuous family of
complete gradient Yamabe shrinkers on warped products which are not isometric
to the Riemannian product in dimension three and higher.
