On a general SU(3) Toda System

TL;DR

This paper investigates a generalized SU(3) Toda system in two dimensions, establishing the existence of radial solutions that bifurcate from known solutions at specific parameter values, expanding understanding of such nonlinear PDE systems.

Contribution

It introduces new bifurcation results for solutions of the SU(3) Toda system, identifying parameter values where solutions branch off from radial solutions.

Findings

Existence of bifurcating radial solutions at specific parameter values.

Explicit characterization of bifurcation points $\mu=rac{2(2 - n - n^2)}{2 + n + n^2}$.

Extension of solution theory for generalized SU(3) Toda systems.

Abstract

We study the following generalized $SU(3)$ Toda System $$ \left\{\begin{array}{ll} -\Delta u=2e^u+\mu e^v & \hbox{ in }\R^2\\ -\Delta v=2e^v+\mu e^u & \hbox{ in }\R^2\\ \int_{\R^2}e^u<+\infty,\ \int_{\R^2}e^v<+\infty \end{array}\right. $$ where $\mu>-2$. We prove the existence of radial solutions bifurcating from the radial solution $(\log \frac{64}{(2+\mu) (8+|x|^2)^2}, \log \frac{64}{ (2+\mu) (8+|x|^2)^2})$ at the values $\mu=\mu_n=2\frac{2-n-n^2}{2+n+n^2},\ n\in\N $.