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On stable and finite Morse index solutions of the fractional Toda system. (English) Zbl 1455.35284

Summary: 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 \(Q \geq 2\) 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 \(n > 2 s\) and \[ \frac{ {\Gamma} ( \frac{ n}{ 2} ) {\Gamma} ( 1 + s )}{ {\Gamma} ( \frac{ n - 2 s}{ 2} )} \frac{ Q ( Q - 1 )}{ 2} > \frac{ {\Gamma}^2 ( \frac{ n + 2 s}{ 4} )}{ {\Gamma}^2 ( \frac{ n - 2 s}{ 4} )} . \] Here, \( \Gamma\) is the Gamma function. When \(Q = 2\), the above equation is the classical (fractional) Gelfand-Liouville equation.

MSC:

35R11 Fractional partial differential equations
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35B45 A priori estimates in context of PDEs
35B08 Entire solutions to PDEs
37B30 Index theory for dynamical systems, Morse-Conley indices
45K05 Integro-partial differential equations
45G15 Systems of nonlinear integral equations
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