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

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

blows up at time $\infty$, then the blow-up points must be the critical points of $f$.