A Liouville theorem for the fractional Ginzburg–Landau equation

Yayun Li; Qinghua Chen; Yutian Lei

doi:10.5802/crmath.91

Elliptical Partial Differential Equations

A Liouville theorem for the fractional Ginzburg–Landau equation

Yayun Li ¹; Qinghua Chen ²; Yutian Lei ²

¹School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing, 210023, China
²Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

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

Abstract

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.

Received: 2020-03-18
Revised: 2020-04-27
Accepted: 2020-06-26
Published online: 2020-10-02

DOI: 10.5802/crmath.91

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

Author's affiliations:

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},
     year = {2020},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {6},
     doi = {10.5802/crmath.91},
     language = {en},
}

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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

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