Supercritical Moser–Trudinger inequalities and related elliptic problems

Abstract

Given $\alpha >0$, we establish the following two supercritical Moser–Trudinger inequalities

$$\begin{aligned} \mathop {\sup }\limits _{ u \in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( (\alpha _n + |x|^\alpha ) |u|^{\frac{n}{n-1}} \big ) dx < +\infty \end{aligned}$$

and

$$\begin{aligned} \mathop {\sup }\limits _{ u\in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( \alpha _n |u|^{\frac{n}{n-1} + |x|^\alpha } \big ) dx < +\infty , \end{aligned}$$

where $W^{1,n}_{0,\mathrm{rad}}(B)$ 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 $W^{1,n}_{0,\mathrm{rad}}(B)$, we should emphasize that the above inequalities fail in $W^{1,n}_{0}(B)$. 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.