We prove that the zero function is the only solution to a certain degenerate PDE defined in the upper half-plane under some geometric assumptions. This result implies that the Euclidean metric is the only adapted compactification of the standard half-plane model of hyperbolic space when the scalar curvature of the compactified metric has a certain sign. These Liouville-type theorems are expected to handle the boundary curvature blow-up to prove compactness results of CCE(conformally compact Einstein) manifolds with positive scalar curvature on the conformal infinity.