A remark on multiplicity of solutions for the Ginzburg-Landau equation
type: Item
instance of: scholarly article
title: A remark on multiplicity of solutions for the Ginzburg-Landau equation
zbMATH Open document ID: 0928.35034
DOI: 10.1016/S0294-1449(99)80014-6
author: Feng Zhou
author: Qing Zhou
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1999-12-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78465
review text: The authors study the critical points of the Ginzburg-Landau functional \[ E_{\varepsilon}(u) = {1 \over 2} \int_{\Omega} \left\{| \nabla u| ^{2} + {1 \over 2\varepsilon^{2}} (1 - | u| ^{2})^{2}\right\} \] defined on \[ \left\{u \in H^{1}(\Omega,{\mathbb C}): u = g \quad\text{on \(\partial\Omega\)}\right\} , \] where \(\varepsilon>0\) and \(g:\partial\Omega \to {\mathbb C}\) is a given smooth map with \(| g(x)| =1\) for any \(x \in \partial\Omega\). Minimizers of this functional have been studied in the book \textit{F. Bethuel}, \textit{H. Brezis} and \textit{F. Hélein} [Ginzburg-Landau Vortices, Birkhäuser (1994; Zbl 0802.35142)]. The main result asserts that, if \(\text{ deg}(g,\partial\Omega) \geq 2\), then for any sufficiently small \(\varepsilon\) the functional \(E_{\varepsilon}\) admits at least \(\text{ deg}(g,\partial\Omega)+1\) distinct critical points. The existence of three critical points was already proved by \textit{L. Almeida} and \textit{F. Bethuel}, [J. Math. Pures Appl., IX. Sér. 77, No. 1, 1-49 (1998; Zbl 0904.35023)].
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1278762
zbMATH Keywords: Ginzburg-Landau equation
zbMATH Keywords: multiple solutions
zbMATH Keywords: Ljusternik-Schnirelman category
reviewed by: Marco Degiovanni
MaRDI profile type: Publication
cites work: Q4838287
cites work: Q4342634
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OpenAlex ID: W2079604749
description: scientific article; zbMATH DE number 1278762