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

TL;DR

This paper extends a fundamental compactness result to fractional Sobolev spaces, providing a new understanding of the behavior of Palais-Smale sequences in these spaces.

Contribution

It generalizes Struwe's global compactness result to fractional Sobolev spaces using profile decomposition techniques.

Findings

Established a global compactness result for fractional Sobolev spaces.

Connected profile decomposition with Palais-Smale sequence analysis.

Provided a simplified proof leveraging existing decomposition methods.

Abstract

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).