Bubbling location for $F$-harmonic maps and inhomogeneous Landau–Lifshitz equations

Abstract

Let f be a positive smooth function on a closed 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)=\int_Mf|\nabla 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 blow up at some critical points of f . As a consequence, if an inhomogeneous Landau–Lifshitz system, i.e. a solution of u_t=u\times\tau_f(u)+\tau_f(u), u: M\rightarrow S^2, blows up at time \infty , then the blow-up points must be the critical points of f .