  In this paper, we study the asymptotic behavior of radial extremal functions
to an inequality involving Hardy potential and critical Sobolev exponent. Based
on the asymptotic behavior at the origin and the infinity, we shall deduce a
strict inequality between two best constants. Finally, as an application of
this strict inequality, we consider the existence of nontrivial solution of a
quasilinear Brezis-Nirenberg type problem with Hardy potential and critical
Sobolev exponent.
