Multi-bump analysis for Trudinger-Moser nonlinearities I. Quantification and location of concentration points

In this paper, we carefully investigate the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension 2 containing a nonlinearity with Trudinger-Moser growth. A quantification result was obtained by the first author in [15] but many questions were left open Similar questions were also explicitly asked in subsequent papers of Del Pino-Musso-Ruf [12], Malchiodi-Martinazzi [30] and Martinazzi [34]. We answer all of them, proving in particular that the blow-up phenomenon is very restrictive because of the strong interaction between bubbles in this equation. This work will have a sequel, giving results on existence of critical points of the associated functional at all energy levels via degree theory arguments, in the spirit of what was done for the Lionville equation in the beautiful work of Chen-Lin [8].