Critical points of the Moser-Trudinger functional on a disk

On the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional $E(u)=\int_{B_1}(e^{u^2}-1)dx, u\in H^1_0(B_1)$ and its restrictions to $M_Λ:=\{u \in H^1_0(B_1):\|u\|^2_{H^1_0}=Λ\}$ for $Λ>0$. We prove that if a sequence $u_k$ of positive critical points of $E|_{M_{Λ_k}}$ (for some $Λ_k>0$) blows up as $k\to\infty$, then $Λ_k\to 4π$, and $u_k\to 0$ weakly in $H^1_0(B_1)$ and strongly in $C^1_{\loc}(\bar B_1\setminus\{0\})$. Using this we also prove that when $Λ$ is large enough, then $E|_{M_Λ}$ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

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