A Liouville theorem for the fractional Ginzburg-Landau equation

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-\alpha}}dy, \end{equation*} where $u:{\mathbb{R}}^n\to {\mathbb{R}}^k$ with $k\ge 1$ and $1<\alpha <n\mathrm{/}2$. We prove that $u \in L^2($\mathbb{R}$^n) \Rightarrow u \equiv 0$on$$\mathbb{R}$^n$,aslongas$u$ is a bounded and differentiable solution.