THE SELF-DUAL CHERN-SIMONS HIGGS EQUATION ON A COMPACT RIEMANN SURFACE WITH BOUNDARY

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with Neumann boundary condition. We show that the Chern-Simons Higgs equation with parameter lambda > 0 has at least two solutions (u(lambda)(1), u(lambda)(2)) for lambda sufficiently large, such that u(lambda)(1) -> -u(0) almost everywhere as lambda -> +infinity, and that u(lambda)(2) -> -infinity almost everywhere lambda -> infinity, where u(0) is a (negative) Green function on M.