Blow-up Analysis in Dimension 2 and a Sharp Form of Trudinger–Moser Inequality

Abstract This paper deals with an improvement of the Trudinger–Moser inequality associated to the embedding of the standard Sobolev space into Orlicz spaces for Ω a smooth bounded domain in ℝ2. The inequality proved here gives in particular precise informations on a previous result obtained by Lions and can be very useful in the study of lack of compactness of the embedding of into {exp(4πu 2) ∈ L 1(Ω)}. We also provide a general asymptotic analysis for sequences of solutions to elliptic PDE's with critical Sobolev growth which blow up at some point. We obtain in particular a result which is well-known in higher dimensions: the concentration points are located at critical points of the regular part of the Green function of the linear operator involved in the equation.