The asymptotic behavior of Chern-Simons Higgs model on a compact Riemann surface with boundary

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 _λ ¹, u _λ ²) for λ sufficiently large, which satisfy that u _λ ¹ → −u ₀ almost everywhere as λ → ∞, and that u _λ ² → −∞ 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 _λ ² − `(u_l ² )]\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.