  Abelian gauge theories consist of a class of field equations which provide a
model for the interaction between matter and electromagnetic fields. In this
paper we analyze the existence of solitary waves for these theories. We assume
that the lower order term W is positive and we prove the existence of solitary
waves if the coupling between matter and electromagnetic field is small. We
point out that the positiveness assumption on W implies that the energy is
positive: this fact makes these theories more suitable to model physical
phenomena.
