Multibumps analysis in dimension 2: Quantification of blow-up levels

In this article, we describe the asymptotic behavior of sequences of solutions to some semilinear elliptic equations with critical exponential growth in planar domains. We prove, in particular, a result analogous to that of Struwe [12] in higher dimensions and extend the two-dimensional result of Adimurthi and Struwe [3] to arbitrary energies. We thus answer a question explicitly asked in this last article.