Mathematics

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.