*Accepted Paper*

**Inserted:** 6 mar 2012

**Last Updated:** 25 feb 2013

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

**Year:** 2012

**Notes:**

16 pages

**Abstract:**

On the unit disk $B_1\subset \mathbb{R}^2$ we study the Moser-Trudinger functional

$E(u)=\int_{B_1}(e^{u^2}-1)dx,\quad u\in H^1_0(B_1)$

and its restrictions to $M_\Lambda$, where $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 $C^1_{loc}(\overline B_1\setminus\{0\})$.

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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