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-\alpha}}dy, \end{equation*} where u:Rn→Rku: \mathbb{R}^{n} \to \mathbb{R}^{k} with k≄1k \geq 1 and 1<α<n/21<\alpha<n/2. We prove that u∈L2(Rn)⇒u≡0u \in L^2(\mathbb{R}^n) \Rightarrow u \equiv 0 on Rn\mathbb{R}^n, as long as uu is a bounded and differentiable solution.Comment: 7 page

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