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 N1 representing the total string number, and a nonnegative integer N2 representing the total anti-string number. The values of the products aN1,aN2∈[0,∞) 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 aN1 and aN2. This improves previously known results.