Supercritical Moser-Trudinger inequalities and related elliptic problems

Abstract

Given $\alpha >0$, we establish the following two supercritical Moser-Trudinger inequalities \[ \sup\limits_{u \in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( (α_n + |x|^α) |u|^{\frac{n}{n-1}} \big) dx < +\infty \] and \[ \sup\limits_{u\in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( α_n |u|^{\frac{n}{n-1} + |x|^α} \big) dx < +\infty, \] where $W_{0,\mathrm{rad}}^{1,n}(B)$ 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 $W_{0,\mathrm{rad}}^{1,n}(B)$, we should emphasize that the above inequalities fail in $W_{0,\mathrm{rad}}^{1,n}(B)$. 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.

23 pages, 0 figure