# Supercritical Moser-Trudinger inequalities and related elliptic problems

## TL;DR

This paper establishes new supercritical Moser-Trudinger inequalities involving radial functions on balls, explores their sharpness and optimal functions, and applies these results to boundary value problems, advancing understanding of functional inequalities in the supercritical regime.

## Contribution

The paper introduces novel supercritical Moser-Trudinger inequalities for radial functions, investigates their sharpness and existence of extremals, and applies findings to elliptic boundary value problems.

## Key findings

- Established two new supercritical inequalities for radial functions.
- Proved the inequalities fail outside the radial class.
- Applied inequalities to specific boundary value problems.

## Abstract

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