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We extend the Global Compactness result by M. Struwe (Math. Z, 1984) to anyfractional 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 simpledirect consequence of the so-called Profile Decomposition of P. Gerard (ESAIM:Control, Optimisation and Calculus of Variations, 1998).

A Global Compactness type result for Palais-Smale sequences in fractional Sobolev spaces