Quantification of Blow-up Levels for a 2-D Elliptic Equation with Critical Exponential Nonlinearity

We study sequences of solutions of some elliptic PDE’s, on 2-dimensional bounded domain, with critical Moser-Trudinger exponential nonlinearity. We prove that lack of compactness occur only when standard bubbles appear and thus we can quantify the levels of blow-up of the underlying functional. The main difficulties, compared to the higher-dimensional situation (Yamabe-type equations), are that subtracting a bubble to a solution changes the nature of the equation satisfied by it and that one has to avoid degenerate bubbles, which could a priori appear as in the study of Chern-Simon vortices.