The existence of solutions for the gravitational Maxwell gauged O(3) model on compact surfaces

Abstract

We consider an elliptic equation induced from the Maxwell gauged O(3) sigma model coupled with gravity. In particular, we study the main equation as two cases: one is for only string and the other is for anti-string. On the compact surface, we prove the existence of ɛ-dependent solutions for each case by using the super-sub solutions method. Moreover, we find the second solution by using the Leray–Schauder degree theory. Furthermore, we estimate the asymptotic behavior of our solution as ɛ → 0.