# Critical points of the Moser-Trudinger functional on a disk

Andrea Malchiodi; Luca Martinazzi

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 5, page 893-908
- ISSN: 1435-9855

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topMalchiodi, 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 -

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