论文标题
Ginzburg-Landau类型的积分方程的Liouville定理
A Liouville theorem for an integral equation of the Ginzburg-Landau type
论文作者
论文摘要
在本文中，我们关注的是非线性积分方程的liouville型结果\ begin {equation*} u（x）= \ oftrightArrow {l}+c _*\ int _ {\ mathbb {r}^{n}}} \ frac {u（1- | U |^{2}}} {| x-y |^y |^{n-α}}} \ end {equation*}在这里$ u：\ mathbb {r}^{n} \ to \ mathbb {r}^{k} $是一个有限的，均匀的，连续的，可区分的功能，具有$ k \ geq 1 $和$ 1 <α<α<n $，$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ roverRightArlOWROW {向量和$ c _*$是一个真正的常数。如果$ u $是有限的能源解决方案，我们证明$ | \ oferrightarrow {l} | \ in \ {0,1 \} $。此外，我们还提供了一个liouville type定理（即$ u \ equiv \ oferrightarrow {l} $）。
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 $k \geq 1$ and $1<α<n$, $\overrightarrow{l} \in \mathbb{R}^{k}$ is a constant vector, and $C_*$ 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., $u \equiv \overrightarrow{l}$).
