A Global Compactness type result for Palais-Smale sequences in   fractional Sobolev spaces
  We extend the Global Compactness result by M. Struwe (Math. Z, 1984) to any fractional Sobolev spaces $\dot{H}^s(\Omega)$ for $0<s<N/2$ and $\Omega \subset \mathbb{R}^N$ a bounded domain with smooth boundary. The proof is a simple direct consequence of the so-called Profile Decomposition of P. Gerard (ESAIM: Control, Optimisation and Calculus of Variations, 1998). 
Giampiero Palatucci, Adriano Pisante
math.AP