*Accepted Paper*

**Inserted:** 6 mar 2012

**Last Updated:** 17 jul 2018

**Journal:** J. Eur. Math. Soc. (JEMS)

**Year:** 2012

16 pages

**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.

**Keywords:**
critical points, Variational methods, Moser-Trudinger inequality, blow-up analysis

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