Existence of solutions to the self-dual equations in the Maxwell gauged O(3) sigma model

In this paper, we prove the existence of solutions in the self-dual Maxwell gauged O(3) sigma model by the super and subsolution method. The supersolutions are explicitly constructed and the subsolutions are given in different ways according to the parameter β representing the decay rates of solutions at infinity. When β has noncritical values, we obtain subsolutions by solving Laplace equations on weighted Sobolev spaces. If β is the critical number, we find the subsolutions via induction and shooting arguments.