Calculus of Variations and Geometric Measure Theory

A. Malchiodi - L. Martinazzi

Critical points of the Moser-Trudinger functional on a disk

created by martinazz on 06 Mar 2012
modified on 17 Jul 2018


Accepted Paper

Inserted: 6 mar 2012
Last Updated: 17 jul 2018

Journal: J. Eur. Math. Soc. (JEMS)
Year: 2012

ArXiv: 1203.1077 PDF

16 pages


On the 2-dimensional unit disk $B_1$ we study the Moser-Trudinger functional $$E(u)=\int{B1}(e{u2}-1)dx, u\in H10(B1)$$ and its restrictions to $M_\Lambda:=\{u \in H^1_0(B_1):\
^2_{H^1_0}=\Lambda\}$ 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