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 . > 0 has at least two solutions (u 1 . , u 2 . ) for . sufficiently large, which satisfy that u 1 . . .u 0 almost everywhere as . . ., and that u 2 . . .. almost everywhere as . . ., where u 0 is a (negative) Green function on M. In this paper, we study the asymptotic behavior of the solutions as . . ., and prove that u 2 . . u2...... 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.