Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources

Details Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources

2011-11-02

arxiv.org/abs/1111.0390v1

Mathematics

Analysis of PDEs

28 pages

Scientific paper

We consider the following Toda system \Delta u_i + \D \sum_{j = 1}^n a_{ij}e^{u_j} = 4\pi\gamma_{i}\delta_{0} \text{in}\mathbb R^2, \int_{\mathbb R^2}e^{u_i} dx < \infty, \forall 1\leq i \leq n, where $\gamma_{i} > -1$, $\delta_0$ is Dirac measure at 0, and the coefficients $a_{ij}$ form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result: $$\sum_{j=1}^n a_{ij}\int_{\R^2}e^{u_j} dx = 4\pi (2+\gamma_i+\gamma_{n+1-i}), \;\;\forall\; 1\leq i \leq n.$$ This generalizes the classification result by Jost and Wang for $\gamma_i=0$, $\forall \;1\leq i\leq n$. (ii) We prove that if $\gamma_i+\gamma_{i+1}+...+\gamma_j \notin \mathbb Z$ for all $1\leq i\leq j\leq n$, then any solution $u_i$ is \textit{radially symmetric} w.r.t. 0. (iii) We prove that the linearized equation at any solution is \textit{non-degenerate}. These are fundamental results in order to understand the bubbling behavior of the Toda system.

Affiliated with

Also associated with

No associations

Rating

Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-327544