Critical points of the Moser-Trudinger functional on a disk

Abstract

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

and its restrictions $E{|}_{M_{\Lambda }}$, where $M_{\Lambda }:=\{u\in H_0^1(B_1):\Vert u{\Vert }_{H_0^1}^2=\Lambda \}$ 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 $C_{\mathrm{loc}}^1({\overline{B}}_1\setminus \{0\})$. Using this fact 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.