A Liouville theorem for the fractional Ginzburg-Landau equation

Abstract

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-α}}dy, \end{equation*} where $u:{\mathbb{R}}^n\to {\mathbb{R}}^k$ with $k\ge 1$ and $1<\alpha <n/2$. We prove that $u\in L^2({\mathbb{R}}^n)\Rightarrow u\equiv 0$ on ${\mathbb{R}}^n$, as long as $u$ is a bounded and differentiable solution.

7 pages

References in corpus (2)

• Fractional generalization of the Ginzburg-Landau equation: An unconventional approach to critical phenomena in complex media
• Sharp criteria of Liouville type for some nonlinear systems

Cited by in corpus (1)

• A Liouville theorem for an integral equation of the Ginzburg-Landau type