Supercritical Moser-Trudinger inequalities and related elliptic problems

Given $\alpha >0$, we establish the following two supercritical Moser-Trudinger inequalities $$\underset{u\in W_{0,\mathrm{r}\mathrm{a}\mathrm{d}}^{1,n}(B):{\int }_B\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{n}dx\le 1}{\sup }{\int }_B\exp (({\alpha }_n+\mathrm{\mid }x{\mathrm{\mid }}^{\alpha })\mathrm{\mid }u{\mathrm{\mid }}^{\frac{n}{n-1}})dx<+\mathrm{\infty }$$ and $$\underset{u\in W_{0,\mathrm{r}\mathrm{a}\mathrm{d}}^{1,n}(B):{\int }_B\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{n}dx\le 1}{\sup }{\int }_B\exp \left({\alpha }_n\mathrm{\mid }u{\mathrm{\mid }}^{\frac{n}{n-1}+\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}\right)dx<+\mathrm{\infty },$$ where $W_{0,\mathrm{r}\mathrm{a}\mathrm{d}}^{1,n}(B)$ is the usual Sobolev spaces of radially symmetric functions on $B$ in ${\mathbb{R}}^n$ with $n\ge 2$. Without restricting to the class of functions $W_{0,\mathrm{r}\mathrm{a}\mathrm{d}}^{1,n}(B)$, we should emphasize that the above inequalities fail in $W_{0,\mathrm{r}\mathrm{a}\mathrm{d}}^{1,n}(B)$. 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.