01111nas a2200121   4500008004100000245006400041210006300105260003400168520070800202100002200910700002100932856003600953       2014                        en  d00aCritical points of the Moser-Trudinger functional on a disk0 aCritical points of the MoserTrudinger functional on a disk  bEuropean Mathematical Society3 aOn 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.1 aMalchiodi, Andrea1 aMartinazzi, Luca  uhttp://hdl.handle.net/1963/6560