  A Sobolev type embedding for radially symmetric functions on the unit ball
$B$ in $\mathbb R^n$, $n\geq 3$, into the variable exponent Lebesgue space
$L_{2^\star + |x|^\alpha} (B)$, $2^\star = 2n/(n-2)$, $\alpha>0$, is known due
to J.M. do \'O, B. Ruf, and P. Ubilla, namely, the inequality \[
  \sup\Big\{\int_B |u(x)|^{2^\star+|x|^\alpha} dx : u\in H^1_{0,{\rm rad}}(B),
\|\nabla u\|_{L^2(B)} =1\Big\} < +\infty
  \] holds. In this work, we generalize the above inequality for higher order
Sobolev spaces of radially symmetric functions on $B$, namely, the embedding \[
H^m_{0,{\rm rad}}(B) \hookrightarrow L_{2_m^\star + |x|^\alpha} (B) \] with
$2\leq m < n/2$, $2_m^* = 2n/(n-2m)$, and $\alpha>0$ holds. Questions
concerning the sharp constant for the inequality including the existence of the
optimal functions are also studied. To illustrate the finding, an application
to a boundary value problem on balls driven by polyharmonic operators is
presented. This is the first in a set of our works concerning functional
inequalities in the supercritical regime.
