Existence of a fully invariant critical functional for the Trudinger–Moser nonlinearity

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 dilation defined in the paper. A functional is fully invariant if it is invariant under these disk automorphisms and radial dilations. The Trudinger–Moser functional is the critical exponential functional on $H_0^1({\mathbb D})$. Existence conjecture. There is a continuous convex functional $J(u)$ on $H_0^1({\mathbb D})$, bounded for $\|\nabla u\|_2\leq 1$, satisfying

• $J(u\circ\eta_\zeta)=J(u)$ for all $\zeta\in{\mathbb D}$;
• $J(u\circ h_s)=J(u)$ for all $s>0$ and radial $u$;
• $J$ lacks weak continuity at any point; and
• $J$ induces an Orlicz space in which the Trudinger–Moser functional is continuous and bounded on every bounded set.

The proposed functional would provide a genuinely critical nonlinearity in the two-dimensional Sobolev space while retaining the natural conformal and radial dilation invariances. The preceding discussion rules out nontrivial functionals of the simpler form $\int_{\mathbb D}F(|x|,u)\,dx$ with both invariance properties, so the conjecture concerns a more general functional.