Year: 2024
Type: article
Abstract: 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].
DOI: 10.3934/dcds.2024155
Authors: Mostafa Fazly, Juncheng Wei, Wen Yang
Institutions: Chinese University of Hong Kong, University of Macau
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4404707056