Lack of compactness in the 2D critical Sobolev embedding, the general case

This paper is devoted to the description of the lack of compactness of the Sobolev embedding of H1(R2) in the critical Orlicz space L(R2). It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in Moser (1971) [44], as in the radial setting investigated in Bahouri et al. (2011) [12]. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an L∞ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of H1(R2), the strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations. The essential ingredient to extract cores consists in proving by contradiction that if the mass responsible for the lack of compactness of the Sobolev embedding in the Orlicz space is scattered, then the energy used would exceed that of the starting sequence.

RésuméCet article est consacré à la description du défaut de compacité de l'inclusion de Sobolev de H1(R2) dans l'espace d'Orlicz critique L(R2). On démontre que la description donnée dans Bahouri et al. (2011) [12] concernant le cas radial reste valable dans le cas général (à des translations près par des cœurs de concentration). La démonstration utilise des arguments de capacité ainsi qu'un processus d'extraction de concentrations.