Classification of solutions of elliptic equations arising from a gravitational O(3) gauge field model

In this paper, we study an elliptic equation arising from the self-dual Maxwell gauged $O(3)$ sigma model coupled with gravity. When the parameter τ equals 1 and there is only one singular source, we consider radially symmetric solutions. There appear three important constants: a positive parameter a representing a scaled gravitational constant, a nonnegative integer $N_1$ representing the total string number, and a nonnegative integer $N_2$ representing the total anti-string number. The values of the products $a{N}_1,a{N}_2\in 0,\infty )$ play a crucial role in classifying radial solutions. By using the decay rates of solutions at infinity, we provide a complete classification of solutions for all possible values of $a{N}_1$ and $a{N}_2$. This improves previously known results.