SU(3) Chern–Simons Vortex Theory and Toda Systems

Motivated by the study of the asymptotic properties of “non-topological” condensates in the non-abelian Chern–Simons vortex theory (see [26]), we analyze the SU(3) Toda system: where M= 2/ 2, K=(kij)= is the SU(3) Cartan matrix and λj are positive parameters. We study the variational problem associated with the system (P)λ1,λ2 in a range of parameters, where the trivial solution is a strict local minimum and the corresponding Sobolev-type inequality fails to apply. In this situation, a lack of compactness may occur due to concentration phenomena. Nonetheless, we are able to establish the existence of a non-trivial solution for (P)λ1,λ2 which is not a minimizer