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

TL;DR

This paper characterizes the lack of compactness in the critical Sobolev embedding of H^1(R^2) into an Orlicz space, using capacity arguments and concentration profiles inspired by Moser's examples, extending previous radial results.

Contribution

It introduces a new approach based on capacity arguments to describe the concentration phenomena in the non-radial setting, differing from previous radial analyses.

Findings

Profile decomposition in terms of Moser concentration profiles.

Core concentration analysis using capacity arguments.

Contradiction-based proof for mass concentration phenomena.

Abstract

This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space ${\cL}(\R^2)$. It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in \cite{M} as in the radial setting investigated in \cite{BMM}. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an $L^ \infty$ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of $H^1(\R^2)$, 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.