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

Fazly, Mostafa; Wei, Juncheng; Yang, Wen

We classify finite Morse index solutions of the following Gelfand-Liouville equation$ \begin{equation*} (-\Delta)^{s} u = e^u \ \ \text{in} \ \ \mathbb{R}^n, \end{equation*} $for $ 12s $ and$ \begin{equation*} \label{1.condition} \frac{ \Gamma^2(\frac{n 2s}{4}) }{ \Gamma^2 (\frac{n-2s}{4})} < \frac{\Gamma(\frac{n}{2}) \Gamma(1 s)}{ \Gamma(\frac{n-2s}{2})}, \end{equation*} $where $ \Gamma $ is the classical Gamma function. The cases of $ s = 1 $ and $ s = 2 $ are settled by Dancer and Farina [12,11] and Dupaigne et al. [16], respectively, using Moser iteration arguments established by Crandall and Rabinowitz [10]. The case of $ 0<s<1 $ is established by Hyder-Yang in [31] applying arguments provided in [14].

GAMMA functions; CLASSIFICATION; A priori; MORSE theory; ARGUMENT; DANCERS

Discrete & Continuous Dynamical Systems: Series A, 2025, Vol 45, Issue 6, p1

1078-0947

Academic Journal

10.3934/dcds.2024155