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

Abstract

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\overrightarrow{l}+C_*\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-α}}dy. \end{equation*} Here $u:{\mathbb{R}}^n\to {\mathbb{R}}^k$ is a bounded, uniformly continuous and differentiable function with $k\ge 1$ and $1<\alpha <n$, $\overrightarrow{l}\in {\mathbb{R}}^k$ is a constant vector, and $C_{\ast }$ is a real constant. If $u$ is the finite energy solution, we prove that $|\overrightarrow{l}|\in \{0,1\}$. Furthermore, we also give a Liouville type theorem (i.e., $u\equiv \overrightarrow{l}$).

16 pages. arXiv admin note: text overlap with arXiv:2006.12664

References in corpus (2)

• Fractional generalization of the Ginzburg-Landau equation: An unconventional approach to critical phenomena in complex media
• A Liouville theorem for the fractional Ginzburg-Landau equation