The paper studies continuity properties of the functional in the Moser-Trudinger inequality on the Sobolev space \(W^{1,N}\) of domains (not necessarily bounded) in \(\mathbb {R}^{N}\) . Given that the functional is not weakly continuous, one may describe its asymptotic properties on bounded sequences in \(W^{1,N}\) in terms of concentration compactness and, more specifically, in the terms of a profile decomposition. While the well-known profile decomposition in \(W^{1,p}(\mathbb {R}^{N})\) expresses the concentration as a sum of elementary concentrations (“bubbles”) of the form \(t_{k}^{\frac{N-p}{p}}w(t(x-y_{k}))\) , elementary concentrations in the case \(p=N\) have the form \(s_{k}^{\frac{N-1}{N}}w(|x-y_{k}|^{1/s_{k}})\) , \(s_{k}\rightarrow \infty \) , with always radial profiles w. We also prove that nonlinear functional in the Moser-Trudinger inequality fails to be weakly continuous only on exceptional sequences.
