Critical points of the Moser-Trudinger functional on a disk

Malchiodi, Andrea, and Martinazzi, Luca. "Critical points of the Moser-Trudinger functional on a disk." Journal of the European Mathematical Society 016.5 (2014): 893-908. <http://eudml.org/doc/277245>.

@article{Malchiodi2014,
abstract = {On the unit disk $B_1\subset \mathbb \{R\}^\{2\}$ we study the Moser-Trudinger functional $E(u)=\int _\{B_1\}\Big (e^\{u^2\}-1\Big )dx,\quad u\in H^1_0(B_1)$ and its restrictions $E|_\{M_\Lambda \}$, where $M_\{\Lambda \}:=\lbrace u\in H^1_0(B_1):\Vert u\Vert ^2_\{H^1_0\}=\Lambda \rbrace $ 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\rightarrow \infty $, then $\Lambda _k\rightarrow 4\pi $, and $u_k\rightarrow 0$ weakly in $H^1_0(B_1)$ and strongly in $C^1_\{\mathrm \{loc\}\}(\overline\{B\}_1\setminus \lbrace 0\rbrace )$. 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.},
author = {Malchiodi, Andrea, Martinazzi, Luca},
journal = {Journal of the European Mathematical Society},
keywords = {Moser-Trudinger inequality; critical points; blow-up analysis; variational methods; Moser-Trudinger functional; critical points; blow-up analysis; variational methods},
language = {eng},
number = {5},
pages = {893-908},
publisher = {European Mathematical Society Publishing House},
title = {Critical points of the Moser-Trudinger functional on a disk},
url = {http://eudml.org/doc/277245},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Malchiodi, Andrea
AU - Martinazzi, Luca
TI - Critical points of the Moser-Trudinger functional on a disk
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 5
SP - 893
EP - 908
AB - On the unit disk $B_1\subset \mathbb {R}^{2}$ we study the Moser-Trudinger functional $E(u)=\int _{B_1}\Big (e^{u^2}-1\Big )dx,\quad u\in H^1_0(B_1)$ and its restrictions $E|_{M_\Lambda }$, where $M_{\Lambda }:=\lbrace u\in H^1_0(B_1):\Vert u\Vert ^2_{H^1_0}=\Lambda \rbrace $ 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\rightarrow \infty $, then $\Lambda _k\rightarrow 4\pi $, and $u_k\rightarrow 0$ weakly in $H^1_0(B_1)$ and strongly in $C^1_{\mathrm {loc}}(\overline{B}_1\setminus \lbrace 0\rbrace )$. 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.
LA - eng
KW - Moser-Trudinger inequality; critical points; blow-up analysis; variational methods; Moser-Trudinger functional; critical points; blow-up analysis; variational methods
UR - http://eudml.org/doc/277245
ER -