Critical points of the moser-trudinger functional on a disk

Abstract

On the unit disk Bℝ2 we study the Moser-Trudinger functional E(u)= ∫ Beu2 - 1) dx; u ε H 1 0 (B; and its restrictions E|MΛ, where MΛ (u ε H1 0 (B V kuk2 H1 = Λ)for Λ > 0. We prove that if a sequence uκ of positive critical points of E|MΛκ (for some Λκ > 0) blows up as k → 1, then Λk → 4π, and uκ → 0 weakly in H1 0 (B and strongly in C 1 Bn {0}). Using this fact we also prove that when Λ is large enough, then E|MΛ has no positive critical point, complementing previous existence results by Carleson-Chang, Struwe and Lamm-Robert-Struwe. © European Mathematical Society 2014.