Bubbling solutions for the Chern–Simons O(3) sigma model on a torus

We consider a nonlinear elliptic equation derived from the Chern–Simons gauged (3) sigma model on a flat torus. Recently, it has been conjectured that there might exist a stable nontopological entire solution in ℝ² under certain conditions. In this paper, we construct bubbling stable solutions on a flat torus which blow up at a vortex point. Our result shows that some global condition on the coefficients in Bartolucci et al. (Ann Inst Henri Poincaré Anal Non Linéaire. doi:10.1016/j.anihpc.2014.03.001, 2014) is necessary for the equivalence result between stable solutions and topological solutions. Moreover, we investigate other types of bubbling solutions including mountain pass solutions and analyze the asymptotic behavior of general solutions on a flat torus.