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 $$E(u)={\int }_{B_1}(e^{u^2}-1)dx,u\in H_0^1(B_1)$$ and its restrictions to $M_{\Lambda }:=\{u\in H_0^1(B_1):\Vert u{\Vert }_{H_0^1}^2=\Lambda \}$ 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 $H_0^1(B_1)$ 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.

16 pages

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