Critical points of the Moser-Trudinger functional on a disk

TL;DR

This paper analyzes the behavior of positive critical points of the Moser-Trudinger functional on a disk, showing blow-up behavior at a specific parameter value and establishing the non-existence of positive critical points for large parameters.

Contribution

It characterizes the blow-up behavior of critical points and proves non-existence of positive critical points for large energy levels, extending previous results.

Findings

Blow-up occurs only as the parameter approaches 4π.

Critical points tend to zero weakly in H^1_0 and strongly away from the origin.

No positive critical points exist when the parameter is sufficiently large.

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.