Existence of nontopological solutions of the self-dual Einstein-Maxwell-Higgs equations on compact surfaces

In this paper, we show the existence of nontopological solutions to the self-dual Einstein-Maxwell-Higgs equations on a compact surface when the parameter $${\varepsilon }>0$$ is small enough and the total string number N is bigger than two. This improves the previously known results by removing any extra conditions on N. Based on degree theory and blowup analysis, we introduce a new method to obtain nontopological solutions.