On Toda system with Cartan matrix $G_2$

Abstract:

We consider the Toda system \begin{align*} \Delta u_i + \sum _{j = 1}^2 a_{ij}e^{u_j} = 4\pi \gamma _{i}\delta _{0} \;\; \text {in }\mathbb R^2, \quad \int _{\mathbb R^2}e^{u_i} dx < \infty ,\;\; \mbox { for } i=1,2, \end{align*} where $\gamma _{i} > -1$, $\delta _0$ is the Dirac measure at $0$, and the coefficients $a_{ij}$ are of the Cartan matrix of rank 2: $A_2, B_2(=C_2),G_2$. Previously, the authors have gotten the classification and non-degeneracy results of solutions for Cartan matrix $A_2$ and $B_2$. In this paper, we consider the $G_2$ case, and we completely classify the solutions and obtain the quantization result as well as the non-degeneracy of solutions for the $G_2$ Toda system.