An abstract version of the concentration compactness principle

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On existence of minimizers for the Hardy–Sobolev–Maz’ya inequality (2005, methodology): “Instead, following the method of [17], it uses Lemma 5.”
Minimal-mass blowup solutions of the mass-critical NLS (2006, methodology): “In other words, G is a group of dislocations in the sense of [29], and is thus a suitable group for constructing a concentration-compactness theory.”
The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality (2008, background): “Another important ingredient of the arguments is the idea of the concentration-compactness principle which aims to compensate for the defect of compactness of the Strichartz inequality, and was exploited in [Bahouri and Gérard 1999; Merle and Vega 1998; Carles and Keraani 2007; Bégout and Vargas 200”