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 H¹₀(Ω) 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 H¹₀(Ω) into exp(4πu²)∈ L¹(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.