Critical points of the Moser-Trudinger functional on a disk

Abstract: 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 $M_\Lambda:={u \in H^1_0(B_1):|u|^2_{H^1_0}=\Lambda}$ for $\Lambda>0$. We prove that if a sequence $u_k$ of positive critical points of $E|{M{\Lambda_k}}$ (for some $\Lambda_k>0$) blows up as $k\to\infty$, then $\Lambda_k\to 4\pi$, 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 $\Lambda$ is large enough, then $E|{M\Lambda}$ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.