On compactness in the Trudinger-Moser inequality

TL;DR

This paper investigates the weak continuity properties of the Moser functional in two dimensions, revealing that it generally maintains weak continuity except in specific concentrating sequences, and introduces a structural analysis akin to profile decomposition.

Contribution

It provides a detailed structural description of the defect of weak convergence for the Moser functional, extending profile decomposition techniques to the two-dimensional case.

Findings

Weak continuity of Moser functional generally holds in two dimensions.

Failure of weak continuity occurs only in specific concentrating sequences.

Structural description of the defect of weak convergence is established.

Abstract

The paper studies continutity of Moser nonlinearity in two dimensions with respect to weak convergence. Unlike the critical nonlinearity in the Sobolev inequality, which lacks weak continuity at any point, Moser functional fails to be weakly continuous only in an exceptional case of a concentrating sequence of functions from the Moser family (up to translations and the remainder vanishing in Sobolev norm). The argument is based on a structural description of the defect of weak converegence, analogous to the profile decomposition established by Solimini for Sobolev inequalities, but involving gauge operators specific for the two-dimensional case.