Abstract
We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition. In the previous paper, we show that the Chern-Simons Higgs equation with parameter λ > 0 has at least two solutions (u 1 λ , u 2 λ ) for λ sufficiently large, which satisfy that u 1 λ → −u 0 almost everywhere as λ → ∞, and that u 2 λ → −∞ almost everywhere as λ → ∞, where u 0 is a (negative) Green function on M. In this paper, we study the asymptotic behavior of the solutions as λ → ∞, and prove that u 2 λ − \(\overline {u_\lambda ^2 } \) converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary ∂M is negative, or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.
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Supported by National Natural Science Foundation of China (Grant Nos. 10701064, 10931001) and XINXING Project of Zhejiang University
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Wang, M. The asymptotic behavior of Chern-Simons Higgs model on a compact Riemann surface with boundary. Acta. Math. Sin.-English Ser. 28, 145–170 (2012). https://doi.org/10.1007/s10114-012-9359-0
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DOI: https://doi.org/10.1007/s10114-012-9359-0