A Liouville theorem for the fractional Ginzburg–Landau equation

Li, Yayun; Chen, Qinghua; Lei, Yutian

doi:10.5802/crmath.91

Équations aux dérivées partielles elliptiques

A Liouville theorem for the fractional Ginzburg–Landau equation

Li, Yayun ¹ ; Chen, Qinghua ² ; Lei, Yutian 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

Comptes Rendus. Mathématique, Tome 358 (2020) no. 6, pp. 727-731.

Résumé

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

u (x) = \int_{ℝ^{n}} \frac{u (1 - | u |^{2})}{{| x - y |}^{n - α}} d y,

where $u : ℝ^{n} \to ℝ^{k}$ with $k \geq 1$ and $1 < α < n / 2$ . We prove that $u \in L^{2} (ℝ^{n}) \Rightarrow u \equiv 0$ on $ℝ^{n}$ , as long as $u$ is a bounded and differentiable solution.

Reçu le : 2020-03-18
Révisé le : 2020-04-27
Accepté le : 2020-06-26
Publié le : 2020-10-02

DOI : 10.5802/crmath.91

Classification : 45G05, 45E10, 35Q56, 35R11

Affiliations des auteurs :

Li, Yayun ¹ ; Chen, Qinghua ² ; Lei, Yutian ²

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     author = {Li, Yayun and Chen, Qinghua and Lei, Yutian},
     title = {A {Liouville} theorem for the fractional {Ginzburg{\textendash}Landau} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {727--731},
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     url = {http://www.numdam.org/articles/10.5802/crmath.91/}
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Li, Yayun; Chen, Qinghua; Lei, Yutian. A Liouville theorem for the fractional Ginzburg–Landau equation. Comptes Rendus. Mathématique, Tome 358 (2020) no. 6, pp. 727-731. doi : 10.5802/crmath.91. http://www.numdam.org/articles/10.5802/crmath.91/

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