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

Motivated by the study of the asymptotic properties of ``non-topological'' condensates in the nonabelian Chern-Simons vortex theory (see \cite{NolascoTarantello3}), we analyze the $SU(3)$ Toda system: \begin{equation*} (P)_{\lambda_{1},\lambda_{2}} \begin{cases} - \Delta z_{j} = \lambda_{j} \left( \displaystyle{ \frac{\text{e}^{\sum_{i=1}^{2} k_{ij} z_{i}}}{\int_{M} \text{e}^{\sum_{i=1}^{2} k_{ij} z_{i}}} } - 1 \right) \qquad \text{on} \ M\\ \int_{M} z_{j} =0 \qquad j=1,2 ;\\ \end{cases} \end{equation*} where $M= \R^{2} / \Z^{2}$, $K=(k_{ij}) = \left(\begin{matrix} 2 & -1 \\ -1 & 2 \end{matrix} \right)$ is the $SU(3)$ Cartan matrix and $\lambda_{j}$ are positive parameters. We study the variational problem associated to the system $(P)_{\lambda_{1},\lambda_{2}}$ 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 nontrivial solution for $(P)_{\lambda_{1},\lambda_{2}} $ which is not a minimizer.