In this paper, we introduce the notion of weakly
-stationary map with potential which is a critical point of the functional
with respect to variations in the domain. It is a generalization of
-stationary maps with potential. We obtain some Liouville theorems for these maps under some curvature conditions of the domain manifolds and some conditions on
. We obtain similar theorems for maps obeying a class of integral equations involving the stress-energy tensor.