Let be a positive smooth function on a closed Riemann surface . The -energy of a map from to a Riemannian manifold is defined as
In this paper, we will study the blow-up properties of Palais--Smale sequences for . We will show that, if a Palais--Smale sequence is not compact, then it must blow up at some critical points of . As a consequence, if an inhomogeneous Landau--Lifshitz system, i.e. a solution of
blows up at time , then the blow-up points must be the critical points of .