We develop a monotonicity formula for solutions of the fractional Toda system $$ (-\Delta)^s f_\alpha = e^{-(f_{\alpha+1}-f_\alpha)} - e^{-(f_\alpha-f_{\alpha-1})} \quad \text{in} \ \ \mathbb R^n,$$ when $0 2s$ and $$ \dfrac{\Gamma(\frac{n}{2})\Gamma(1+s)}{\Gamma(\frac{n-2s}{2})} \frac{Q(Q-1)}{2} > \frac{ \Gamma(\frac{n+2s}{4})^2 }{ \Gamma(\frac{n-2s}{4})^2} . $$ Here, $\Gamma$ is the Gamma function. When $Q=2$, the above equation is the classical (fractional) Gelfand-Liouville equation.

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关于分数阶 Toda 系统的稳定有限莫尔斯指数解

我们为分数 Toda 系统 $$ (-\Delta)^s f_\alpha = e^{-(f_{\alpha+1}-f_\alpha)} - e^{-(f_ \alpha-f_{\alpha-1})} \quad \text{in} \ \ \mathbb R^n,$$ 当 $0 2s$ 和 $$ \dfrac{\Gamma(\frac{n}{2} )\Gamma(1+s)}{\Gamma(\frac{n-2s}{2})} \frac{Q(Q-1)}{2} > \frac{ \Gamma(\frac{n+ 2s}{4})^2 }{ \Gamma(\frac{n-2s}{4})^2} 。$$ 这里，$\Gamma$ 是Gamma 函数。当 $Q=2$ 时，上述方程为经典（分数）Gelfand-Liouville 方程。