# Sharp threshold nonlinearity for maximizing the Trudinger-Moser inequalities

## TL;DR

This paper investigates the precise conditions under which maximizers exist for the Trudinger-Moser inequality with critical growth nonlinearities on the plane and disk, providing explicit sharp thresholds and asymptotic expansions.

## Contribution

It introduces a sharp threshold nonlinearity that delineates existence and non-existence of maximizers, with explicit asymptotic formulas involving Apéry's constant.

## Key findings

- Derived explicit sharp threshold nonlinearity for maximizer existence
- Provided asymptotic expansions for critical growth nonlinearities
- Extended results to exponential radial Sobolev inequalities on R^2

## Abstract

We study existence of maximizer for the Trudinger-Moser inequality with general nonlinearity of the critical growth on $R^2$, as well as on the disk. We derive a very sharp threshold nonlinearity between the existence and the non-existence in each case, in asymptotic expansions with respect to growth and decay of the function. The expansions are explicit, using Ap\'ery's constant. We also obtain an asymptotic expansion for the exponential radial Sobolev inequality on $R^2$.