An existence theorem for generalized Abelian Higgs equations and its application

In this note, we construct self-dual vortices and cosmic strings from the generalized Abelian Higgs theory in which the Higgs potential is a polynomial whose degree depends on the number m . When | m | > 0 , we obtain sharp existence theorems for vortices and strings corresponding to the absence and presence of gravity, respectively, over the whole plane. In the absence of gravity, we prove the existence of vortices by using a monotone iteration method. When gravity is taken into account, a regularization method and a fixed-point theorem are used to show that multiple string solutions exist under a sufficient condition imposed only on the total number of strings. In addition, a series of properties with respect to vortices and strings have also been established.