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

Abstract

We develop a monotonicity formula for solutions of the fractional Toda system $$(-\Delta {)}^s{f}_{\alpha }=e^{-(f_{\alpha +1}-f_{\alpha })}-e^{-(f_{\alpha }-f_{\alpha -1})}\text{in}{\mathbb{R}}^n,$$ when $0<s<1$, $\alpha =1,\cdots ,Q$, $f_0=-\infty$, $f_{Q+1}=\infty$ and $Q\ge 2$ 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 $$\frac{\Gamma (\frac{n}{2})\Gamma (1+s)}{\Gamma (\frac{n-2s}{2})}\frac{Q(Q-1)}{2}>\frac{\Gamma (\frac{n+2s}{4}{)}^2}{\Gamma (\frac{n-2s}{4}{)}^2}.$$ Here, $\Gamma$ is the Gamma function. When $Q=2$, the above equation is the classical (fractional) Gelfand-Liouville equation.

21 pages. Comments are welcome

References in corpus (1)

• Liouville theorems and $1$-dimensional symmetry for solutions of an elliptic system modelling phase separation

Cited by in corpus (1)

• On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand-Liouville equation