Existence of Dyons in the Coupled Georgi-Glashow-Skyrme Model

We prove the existence of a continuous family of finite-energy particle-like solutions in the coupled Georgi-Glashow-Skyrme model carrying both electric and magnetic charges, known as dyons. Due to the presence of electricity and the Minkowski spacetime signature, we need to solve a variational problem with an indefinite action functional. Our results show that, while the magnetic charge is uniquely determined by the topological monopole number, the electric charge of a solution can be arbitrarily prescribed in an open interval.