대한수학회지 (Journal of the Korean Mathematical Society)

  • 제58권6호
  • /
  • Pages.1327-1345
  • /
  • 2021
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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A LIOUVILLE THEOREM OF AN INTEGRAL EQUATION OF THE CHERN-SIMONS-HIGGS TYPE

  • Chen, Qinghua (Institute of Mathematics School of Mathematical Sciences Nanjing Normal University) ;
  • Li, Yayun (School of Applied Mathematics Nanjing University of Finance & Economics) ;
  • Ma, Mengfan (Institute of Mathematics School of Mathematical Sciences Nanjing Normal University)
  • 투고 : 2020.11.09
  • 심사 : 2021.07.29
  • 발행 : 2021.11.01
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초록

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation of Chern-Simons-Higgs type $$u(x)=\vec{\;l\;}+C_{\ast}{{\displaystyle\smashmargin{2}{\int\nolimits_{\mathbb{R}^n}}}\;{\frac{(1-{\mid}u(y){\mid}^2){\mid}u(y){\mid}^2u(y)-\frac{1}{2}(1-{\mid}u(y){\mid}^2)^2u(y)}{{\mid}x-y{\mid}^{n-{\alpha}}}}dy.$$ Here u : ℝn → ℝk is a bounded, uniformly continuous function with k ⩾ 1 and 0 < α < n, $\vec{\;l\;}{\in}\mathbb{R}^k$ is a constant vector, and C* is a real constant. We prove that ${\mid}\vec{\;l\;}{\mid}{\in}\{0,\frac{\sqrt{3}}{3},1\}$ 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 $u{\rightarrow}\vec{\;l\;}$ with ${\mid}\vec{\;l\;}{\mid}=\frac{\sqrt{3}}{3}$, when |x| → ∞, or $u{\equiv}\vec{\;l\;}$, where ${\mid}\vec{\;l\;}{\mid}{\in}\{0,1\}$.

키워드

과제정보

The authors are grateful to the referees for their valuable comments. Their suggestions have improved this article. This research was supported by National Natural Science Foundation of China (11871278).

참고문헌

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