On the lack of compactness in the 2D critical Sobolev embedding

TL;DR

This paper characterizes the lack of compactness in the 2D critical Sobolev embedding into Orlicz spaces, using concentration examples and exploring implications for nonlinear wave equations with exponential growth.

Contribution

It introduces a novel approach to describe the lack of compactness in Sobolev embeddings into Orlicz spaces, distinct from traditional Lebesgue space methods.

Findings

Describes the lack of compactness via concentration examples

Connects the Orlicz norm to solutions of nonlinear wave equations

Provides a new variational perspective on Sobolev embeddings

Abstract

This paper is devoted to the description of the lack of compactness of $H^1_{rad}(\R^2)$ in the Orlicz space. Our result is expressed in terms of the concentration-type examples derived by P. -L. Lions. The approach that we adopt to establish this characterization is completely different from the methods used in the study of the lack of compactness of Sobolev embedding in Lebesgue spaces and take into account the variational aspect of Orlicz spaces. We also investigate the feature of the solutions of non linear wave equation with exponential growth, where the Orlicz norm plays a decisive role.