Nontopological solutions in the self-dual Maxwell–Chern–Simons gauged O(3) sigma model

In this paper, we prove the existence of nontopological solutions in the self-dual Maxwell–Chern–Simons gauged O(3) sigma model in the plane. If the scaling parameter is small enough, the reduced equation can be regarded as a perturbation of the Liouville equation. Then, we use the Generalized Implicit Function Theorem to prove the existence of solutions with nontopological boundary conditions. Using this result, we show the existence of symmetrically topological solutions and nontopological solutions for the original self-dual equations whose magnetic fluxes are slightly bigger than the critical value. Moreover, we also compute the topological charge and the degree of the spin vectors.