Paper detail

On stable and finite Morse index solutions of the fractional Toda system

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.