TY - JOUR T1 - Critical points of the Moser-Trudinger functional on a disk Y1 - 2014 A1 - Andrea Malchiodi A1 - Luca Martinazzi AB - 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. PB - European Mathematical Society UR - http://hdl.handle.net/1963/6560 N1 - 16 pages U1 - 6487 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - RPRT T1 - Critical points of the Moser-Trudinger functional Y1 - 2011 A1 - Francesca De Marchis A1 - Andrea Malchiodi A1 - Luca Martinazzi KW - Moser-Trudinger inequality PB - SISSA UR - http://hdl.handle.net/1963/4592 U1 - 4353 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER -