Non-Abelian Chern–Simons–Higgs system with indefinite functional

In this paper, we are concerned with the general non-Abelian Chern–Simons–Higgs models of rank two. The corresponding self-dual equations can be reduced to a nonlinear elliptic system, and the form is determined by a non-degenerate matrix K. One of the major questions is how the matrix K affects the structure of the solutions to the self-dual equations. There have been some existence results of the solutions to the self-dual equations when det (K) > 0. However, the solvability for the case det (K) < 0 is not fully understood in spite of its physical importance. In contrast to det (K) > 0 , one major difficulty for the case det (K) < 0 is that the energy functional associated with the elliptic system is usually indefinite. The direct variational method thus fails. We overcome this obstacle and obtain a partially positive answer for the solvability when det (K) < 0 by controlling the indefinite functional with a suitable constraint.