论文标题
关于分数TODA系统的稳定和有限的摩尔斯索引解决方案
On stable and finite Morse index solutions of the fractional Toda system
论文作者
论文摘要
我们为分数TODA系统$$（-Δ）^sf_α= e^{ - （f_ {α+1}-f_α）} - E^{ - （f_α-f_ {α-1}} $α= 1，\ cdots，q $，$ f_0 = - \ infty $，$ f_ {q+1} = \ infty $和$ q \ ge2 $是该系统中的方程数。然后，我们应用此公式，技术积分估算，稳定均质解决方案的分类以及吹入分析论点来建立上述系统的有限摩尔斯指数（和稳定）解决方案的Liouville Type定理，当\ dfrac {γ（\ frac {n} {2}）γ（1+s）}} {γ（\ frac {n-2s} {2} {2} {2} {2}} \ frac {q（q-1）} {2} {2} {2}> \ frac {> \ frac {γ（ γ（\ frac {n-2s} {4}）^2}。 $$这里，$γ$是伽马功能。当$ q = 2 $时，上述方程是经典的（分数）gelfand-liouville方程式。
We develop a monotonicity formula for solutions of the fractional Toda system $$ (-Δ)^s f_α= e^{-(f_{α+1}-f_α)} - e^{-(f_α-f_{α-1})} \quad \text{in} \ \ \mathbb R^n,$$ when $0<s<1$, $α=1,\cdots,Q$, $f_0=-\infty$, $f_{Q+1}=\infty$ and $Q \ge2$ is the number of equations in this system. We then apply this formula, technical integral estimates, classification of stable homogeneous solutions, and blow-down analysis arguments to establish Liouville type theorems for finite Morse index (and stable) solutions of the above system when $n > 2s$ and $$ \dfrac{Γ(\frac{n}{2})Γ(1+s)}{Γ(\frac{n-2s}{2})} \frac{Q(Q-1)}{2} > \frac{ Γ(\frac{n+2s}{4})^2 }{ Γ(\frac{n-2s}{4})^2} . $$ Here, $Γ$ is the Gamma function. When $Q=2$, the above equation is the classical (fractional) Gelfand-Liouville equation.
