Lack of compactness in the 2D critical Sobolev embedding, the general case (Q5895549)

It is well known that \(H^1 (\mathbb R^2) \hookrightarrow {\mathcal L} (\mathbb R^2)\) is continuous, but not compact, where \({\mathcal L} (\mathbb R^2)\) is the Orlicz space \(L^\Phi (\mathbb R^2)\) based on the generating function \(\Phi (s) = e^{s^2} -1\). The authors study in detail how to understand the lack of compactness (both near infinity and near the origin) based on Moser's extremal functions in the context of so-called profile decompositions.