  We study an optimal M-partition problem for the Yamabe equation on the round
sphere, in the presence of some particular symmetries. We show that there is a
correspondence between solutions to this problem and least-energy sign-changing
symmetric solutions to the Yamabe equation on the sphere with precisely M nodal
domains. The existence of an optimal partition is established through the study
of the limit profiles of least-energy solutions to a weakly coupled competitive
elliptic system on the sphere.
