Comptes Rendus
Elliptical Partial Differential Equations
A Liouville theorem for the fractional Ginzburg–Landau equation
Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 727-731.

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation

u(x)= n u(1-|u| 2 ) |x-y| n-α dy,

where u: n k with k1 and 1<α<n/2. We prove that uL 2 ( n )u0 on n , as long as u is a bounded and differentiable solution.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.91
Classification: 45G05, 45E10, 35Q56, 35R11

Yayun Li 1; Qinghua Chen 2; Yutian Lei 2

1 School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing, 210023, China
2 Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMATH_2020__358_6_727_0,
     author = {Yayun Li and Qinghua Chen and Yutian Lei},
     title = {A {Liouville} theorem for the fractional {Ginzburg{\textendash}Landau} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {727--731},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {6},
     year = {2020},
     doi = {10.5802/crmath.91},
     language = {en},
}
TY  - JOUR
AU  - Yayun Li
AU  - Qinghua Chen
AU  - Yutian Lei
TI  - A Liouville theorem for the fractional Ginzburg–Landau equation
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 727
EP  - 731
VL  - 358
IS  - 6
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.91
LA  - en
ID  - CRMATH_2020__358_6_727_0
ER  - 
%0 Journal Article
%A Yayun Li
%A Qinghua Chen
%A Yutian Lei
%T A Liouville theorem for the fractional Ginzburg–Landau equation
%J Comptes Rendus. Mathématique
%D 2020
%P 727-731
%V 358
%N 6
%I Académie des sciences, Paris
%R 10.5802/crmath.91
%G en
%F CRMATH_2020__358_6_727_0
Yayun Li; Qinghua Chen; Yutian Lei. A Liouville theorem for the fractional Ginzburg–Landau equation. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 727-731. doi : 10.5802/crmath.91. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.91/

[1] Haïm Brezis Comments on two notes by L. Ma and X. Xu, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 5-6, pp. 269-271 | DOI | Zbl

[2] Haïm Brézis; Frank Merle; Tristan Rivière Quantization effects for -Δu=u(1-|u| 2 ) in 2 , Arch. Ration. Mech. Anal., Volume 126 (1994) no. 1, pp. 35-58 | DOI | Zbl

[3] Gabriella Caristi; Lorenzo D’Ambrosio; Enzo Mitidieri Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., Volume 76 (2008), pp. 27-67 | DOI | MR | Zbl

[4] Rose-Marie Hervé; Michel Hervé Quelques proprietes des solutions de l’equation de Ginzburg-Landau sur un ouvert de 2 , Potential Anal., Volume 5 (1996) no. 6, pp. 591-609 | DOI | Zbl

[5] Yutian Lei; Congming Li Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 6, pp. 3277-3315 | MR | Zbl

[6] Li Ma Liouville type theorem and uniform bound for the Lichnerowicz equation and the Ginzburg-Landau equation, C. R. Math. Acad. Sci. Paris, Volume 348 (2010) no. 17-18, pp. 993-996 | MR | Zbl

[7] Li Ma Boundedness of solutions to Ginzburg-Landau fractional Laplacian equation, Int. J. Math., Volume 27 (2016) no. 5, 1650048, 6 pages | MR | Zbl

[8] Vincent Millot; Yannick Sire On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Ration. Mech. Anal., Volume 215 (2015) no. 1, pp. 125-210 | DOI | MR | Zbl

[9] Alexander V. Milovanov; Jens Juul Rasmussen Fractional generalization of the Ginzburg-Landau equation: an unconventional approach to critical phenomena in complex media, Phys. Lett., A, Volume 337 (2005) no. 1-2, pp. 75-80 | DOI | Zbl

[10] Elias M. Stein Singular integrals and differentiability properties of functions, Princeton Mathematical Series, 30, Princeton University Press, 1970 | MR | Zbl

[11] Vasily E. Tarasov; George M. Zaslavsky Fractional Ginzburg–Landau equations for fractal media, Physica A, Volume 354 (2005), pp. 249-261 | DOI

[12] Xingwang Xu Uniqueness theorem for integral equations and its application, J. Funct. Anal., Volume 247 (2007) no. 1, pp. 95-109 | MR | Zbl

Cited by Sources:

Comments - Policy