论文标题
高阶Gelfand-Liouville方程的有限摩尔斯索引解决方案的分类
Classification of finite Morse index solutions of higher-order Gelfand-Liouville equation
论文作者
论文摘要
我们将以下Gelfand-Liouville方程式对有限的Morse索引解决方案进行了分类 \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.我们表明，上述方程式不接受任何有限的莫尔斯索引解决方案，$（-Δ）^{s/2} u $在无穷大时消失了$ n> 2s $和\ begin {equation*} \ label {1.condition} \ frac {γ（ γ（\ frac {n-2s} {4}）^2} <\ frac {γ（\ frac {n} {2} {2} {2}）γ（1+s）}} {γ（\ frac {n-2s} \ end {equation*}其中$γ$是经典的伽马函数。 $ s = 1 $和$ s = 2 $的情况由舞者和法琳娜\ cite {df，d}和dupaigne等人解决。 \ cite {dggw}分别使用Crandall和Rabinowitz \ Cite {Cr}建立的Moser Iteration参数。 $ 0 <s <1 $的情况是由\ cite {hy}在\ cite {ddw，fw}中提供的参数建立的。
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}.
