  Given $\alpha >0$, we establish the following two supercritical
Moser-Trudinger inequalities \[ \sup\limits_{u \in W^{1,n}_{0,{\rm rad}}(B):
\int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( (\alpha_n + |x|^\alpha)
|u|^{\frac{n}{n-1}} \big) dx < +\infty \] and \[ \sup\limits_{u\in
W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big(
\alpha_n |u|^{\frac{n}{n-1} + |x|^\alpha} \big) dx < +\infty, \] where
$W^{1,n}_{0,{\rm rad}}(B)$ is the usual Sobolev spaces of radially symmetric
functions on $B$ in $\mathbb R^n$ with $n\geq 2$. Without restricting to the
class of functions $W^{1,n}_{0,{\rm rad}}(B)$, we should emphasize that the
above inequalities fail in $W^{1,n}_{0,{\rm rad}}(B)$. Questions concerning the
sharpness of the above inequalities as well as the existence of the optimal
functions are also studied. To illustrate the finding, an application to a
class of boundary value problems on balls is presented. This is the second part
in a set of our works concerning functional inequalities in the supercritical
regime.
