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

Senping Luo, Juncheng Wei, Wenming Zou

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-…