Beyond the Trudinger-Moser supremum

Let Ω be a bounded, smooth domain in ℝ ². We consider the functional, in the supercritical Trudinger-Moser regime, i. e. for ∫ _Ω{pipe}▼u{pipe} ²dx > 4π. More precisely, we are looking for critical points of I(u) in the class of functions u ε H ₀ ¹(Ω) such that ∫ _Ω{pipe}▼u{pipe} ²dx = 4π(1+α), for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with ∫ _Ω{pipe}▼u{pipe} ²dx = 4π(1+α) for any bounded domain Ω, 2-peak critical points with ∫ _Ω{pipe}▼u{pipe} ²dx = 8π(1+α) for non-simply connected domains Ω, and k-peak critical points with ∫ _Ω{pipe}▼u{pipe} ²dx = 4kπ(1+α) if Ω is an annulus.