On the equation $p \frac{Γ(\frac{n}{2}-\frac{s}{p-1})Γ(s+\frac{s}{p-1})}{Γ(\frac{s}{p-1})Γ(\frac{n-2s}{2}-\frac{s}{p-1})} =\frac{Γ(\frac{n+2s}{4})^2}{Γ(\frac{n-2s}{4})^2}$

Abstract

The note is aimed at giving a complete characterization of the following equation: $$\displaystyle p\frac{Γ(\frac{n}{2}-\frac{s}{p-1})Γ(s+\frac{s}{p-1})}{Γ(\frac{s}{p-1})Γ(\frac{n-2s}{2}-\frac{s}{p-1})} =\frac{Γ(\frac{n+2s}{4})^2}{Γ(\frac{n-2s}{4})^2}.$$ The method is based on some key transformation and the properties of the Gamma function. Applications to fractional nonlinear Lane-Emden equations will be given.

14 pages