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

Abstract

The note is aimed at giving a complete characterization of the following equation: $$p\frac{\Gamma (\frac{n}{2}-\frac{s}{p-1})\Gamma (s+\frac{s}{p-1})}{\Gamma (\frac{s}{p-1})\Gamma (\frac{n-2s}{2}-\frac{s}{p-1})}=\frac{\Gamma (\frac{n+2s}{4}{)}^2}{\Gamma (\frac{n-2s}{4}{)}^2}.$$ 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