A Liouville theorem of an integral equation of the Chern-Simons-Higgs type
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In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation of Chern-Simons-Higgs type \begin{equation*} u(x)=\overrightarrow{l}+C_{*}\int_{\mathbb{R}^n}\frac{(1-|u(y)|^2)|u(y)|^2u(y)-\frac{1}{2}(1-|u(y)|^2)^2u(y)}{|x-y|^{n-\alpha}}dy. \end{equation*} Here $u:\mathbb{R}^n\rightarrow \mathbb{R}^k$ is a bounded, uniformly continuous function with $k\geqslant1$ and $0<\alpha<n$,$\overrightarrow{l}\in \mathbb{R}^k$ is a constant vector, and $C_{*}$ is a real constant. We prove that $|\overrightarrow{l}|\in \{0,\frac{\sqrt{3}}{3},1\}$ if $u$ is the finite energy solution. Further, if $u$ is also a differentiable solution, then we give a Liouville type theorem, that is either $u\rightarrow\overrightarrow{l}$ with $|\overrightarrow{l}|=\frac{\sqrt{3}}{3}$, when $|x|\rightarrow \infty$, or $u\equiv\overrightarrow{l},$ where $|\overrightarrow{l}|\in \{0,1\}.$
Abstract
In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation of Chern-Simons-Higgs type \begin{equation*} u(x)=\overrightarrow{l}+C_{*}\int_{\mathbb{R}^n}\frac{(1-|u(y)|^2)|u(y)|^2u(y)-\frac{1}{2}(1-|u(y)|^2)^2u(y)}{|x-y|^{n-\alpha}}dy. \end{equation*} Here $u:\mathbb{R}^n\rightarrow \mathbb{R}^k$ is a bounded, uniformly continuous function with $k\geqslant1$ and $0<\alpha<n$,$\overrightarrow{l}\in \mathbb{R}^k$ is a constant vector, and $C_{*}$ is a real constant. We prove that $|\overrightarrow{l}|\in \{0,\frac{\sqrt{3}}{3},1\}$ if $u$ is the finite energy solution. Further, if $u$ is also a differentiable solution, then we give a Liouville type theorem, that is either $u\rightarrow\overrightarrow{l}$ with $|\overrightarrow{l}|=\frac{\sqrt{3}}{3}$, when $|x|\rightarrow \infty$, or $u\equiv\overrightarrow{l},$ where $|\overrightarrow{l}|\in \{0,1\}.$
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