This work is motivated by the observation [see C. H. Taubes, Commun. Math. Phys. 72, 277-292 (1980; Zbl 0451.35101)] that certain self-dual equations for energy-minimizing multivortices in Chern-Simons theory can be reduced to suitable elliptic equations for the logarithmic value of the particle density. Here a particular class of elliptic equations arising in this way is considered on the 2-dimensional torus. The main result of the paper is an existence theorem for non-trivial solutions of the considered equations which is proven with the help of variational techniques.