Explicit optimal constants of two critical Rellich inequalities for radially symmetric functions

Abstract: We consider two critical Rellich inequalities with singularities at both the origin and the boundary in the higher order critical radial Sobolev spaces $W_{0, {\rm rad}}^{k, p}$, where $1< p = \frac{N}{k}$. We give the explicit values of the optimal constants of two critical Rellich inequalities for radially symmetric functions in $W_{0, {\rm rad}}^{k, \frac{N}{k}}$. Furthermore the (non-)attainability of the optimal constants are also discussed.