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.

中文翻译：

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紧致表面上自对偶Einstein-Maxwell-Higgs方程的非拓扑解的存在性

在本文中，我们证明了当参数\（{\ varepsilon}> 0 \）足够小且总字符串数N较大时，紧对表面上自对偶Einstein-Maxwell-Higgs方程的非拓扑解的存在多于两个。通过消除N上的任何额外条件，可以改善先前已知的结果。基于度理论和爆破分析，我们介绍了一种获取非拓扑解的新方法。