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Classification of finite Morse index solutions of higher-order Gelfand-Liouville equation

We classify finite Morse index solutions of the following Gelfand-Liouville equation \begin{equation*} (-Δ)^{s} u= e^u \ \ \text{in} \ \ \mathbb{R}^n, \end{equation*} for $1<s<2$ and $s=2$ via a novel monotonicity formula and technical blow-down analysis. We show that the above equation does not admit any finite Morse index solution with $(-Δ)^{s/2} u$ vanishes at infinity provided $n>2s$ and \begin{equation*} \label{1.condition} \frac{ Γ(\frac{n+2s}{4})^2 }{ Γ(\frac{n-2s}{4})^2} < \frac{Γ(\frac{n}{2}) Γ(1+s)}{ Γ(\frac{n-2s}{2})}, \end{equation*} where $Γ$ is the classical Gamma function. The cases of $s=1$ and $s=2$ are settled by Dancer and Farina \cite{df,d} and Dupaigne et al. \cite{dggw}, respectively, using Moser iteration arguments established by Crandall and Rabinowitz \cite{CR}. The case of $0<s<1$ is established by Hyder-Yang in \cite{hy} applying arguments provided in \cite{ddw,fw}.