Monotonicity of solution to the dark monopole equations in non-Abelian gauge field theory

In this paper, we combine a two-step iterative shooting argument and a fixed-point theorem approach to establish the existence of the solution of the dark monopole equations in a recently formulated non-Abelian gauge field theory by Deglmann and Kneipp and discuss its qualitative properties. In addition, the energy-minimizing solution of the dark monopole equations also satisfies the monotonicity, thus settling an unsettled but characteristic issue left out in an existence theorem regarding such a solution and confirming evidence from earlier numerical simulations.