  In this paper, we consider the smooth map from a Riemannian manifold to the
standard Euclidean space and the p-Ginzburg-Landau energy. Under suitable
curvature conditions on the domain manifold, some Liouville type theorems are
established by assuming either growth conditions of the p-Ginzburg-Landau
energy or an asymptotic condition at the infinity for the maps. In the end of
paper, we obtain the unique constant solution of the constant Dirichlet
boundary value problems on starlike domains.
