The Relativistic non-abelian Chern-Simons Equations

We study ther xr system of nonlinear elliptic equations ,a=1,2,...,r,x∈R ², where λ τ 0 is a constant parameter,K = (K_ab) is the Cartan matrix of a semi-simple Lie algebra, and β_p is the Dirac measure concentrated atp R ². This system of equations arises in the relativistic non-Abelian Chern-Simons theory and may be viewed as a nonintegrable deformation of the integrable Toda system. We establish the existence of a class of solutions known as topological multivortices. The crucial step in our method is the use of the decomposition theorem of Cholesky for positive definite matrices so that a variational principle can be formulated. Research supported in part by the National Science Foundation under grant DMS-9596041