On a result by Carleson-Chang concerning the Trudinger-Moser inequality

In this note we report on joint work with D. G. de Figueiredo and J.M. do O : It has been shown by N. Trudinger and J. Moser that for normalized functi ons u of the Sobolev space W-1,W-N(Omega), where Omega is a bounded domain in R-N, the integral integral (Omega) exp(mu (alpha NN/(N-1)))dx remains un iformly bounded. L. Carleson and A. Chang proved that there exists a corres ponding extremal function in the case that Omega is the unit ball in R N. W e give a new interpretation of this result, showing that whether the suprem um is attained depends on terms of lower order growth, just as in the celeb rated result of H. Brezis - L. Nirenberg for the case of the Sobolev-imbedd ing W-1,W-2(Omega) subset of L2N/(N-2)(Omega), N greater than or equal to 3 . The key ingredient is the construction of an explicit sequence which is m aximizing for the above integral among all normalized " concentrating seque nces".