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

This Note 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 (1971) in [16], as in the radial setting investigated in Bahouri et al. (2011) [5]. 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from the one conducted in the radial case which is based on an L∞ estimate far away from the origin and which is no longer valid in the general frame work. 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.

RésuméCette Note est consacrée à lʼétude du défaut de compacité de lʼinjection de Sobolev de H1(R2) dans lʼespace dʼOrlicz critique L(R2). Nous démontrons que la déscription donnée dans Bahouri et al. (2011) [5] concernant le cas radial reste valable dans le cas général (à des translations près par des coeurs de concentration). La preuve utilise des arguments de capacité ainsi quʼun processus dʼextraction de concentrations.