3 problems

Uniqueness conjecture. If $\alpha=4\pi$ and $\Omega$ is a ball, then the maximizers of the Trudinger–Moser inequality are unique.

Let $\Omega\subset\mathbb{R}^n$ be a ball, and consider the Trudinger–Moser variational problem … A maximizer is a function attaining this supremum. Uniqueness conjecture. Maximize…

Let $H_0^1({\mathbb D})$ be the Sobolev space on the unit disk, let $\eta_\zeta$ denote the disk automorphism associated with $\zeta\in{\mathbb D}$, and let $h_s$ be the radial dil…