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

Abstract

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 $$E_f(u)=\int_Mf|\nabla u|^2dV_g.$$ 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 $$u_t=u\times\tau_f(u)+\tau_f(u),\s u:M\to S^2$$ blows up at time $\infty$, then the blow-up points must be the critical points of $f$.