A ug 2 00 5 Bubbling location for F - harmonic maps and Inhomogeneous Landau - Lifshitz equations ∗

Let f be a positive smooth function on a close Riemann surface (M,g). The f − energy of a map u from M to a Riemannian manifold (N, h) is defined as E f (u) = M f |∇u| 2 dV g. In this paper, we will study the blow-up properties of Palais-Smale sequences for E f. We will show that, if a Palais-Smale sequence is not compact, then it must blows up at some critical points of f. As a sequence, if an inhomogeneous Landau-Lifshitz system, i.e. a solution of u t = u × τ f (u) + τ f (u), u : M → S 2 blows up at time ∞, then the blow-up points must be the critical points of f .