  In this paper, we show that steady or shrinking complete gradient Yamabe
solitons with finite total scalar curvature and non-positive Ricci curvature
are Ricci flat. Moreover, under certain pinching condition for Ricci curvature,
we show that steady or shrinking complete gradient Yamabe solitons with finite
total scalar curvature and non-positive scalar curvature have zero scalar
curvature.
