Supercritical Moser-Trudinger inequalities and related elliptic problems

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.