Critical points of arbitrary energy for the Trudinger-Moser functional in planar domains

TL;DR

This paper proves the existence of positive critical points for the Trudinger-Moser functional in planar domains with arbitrary energies, using topological and analytical methods.

Contribution

It introduces a novel approach combining degree theory, compactness estimates, and topological arguments to establish critical points for the functional.

Findings

Existence of positive critical points for arbitrary energies

Application of degree theory and Poincaré-Hopf theorem in this context

Sharp compactness estimates for the functional

Abstract

Given a smoothly bounded non-contractible domain $\Omega\subset \mathbb{R}^2$, we prove the existence of positive critical points of the Trudinger-Moser embedding for arbitrary Dirichlet energies. This is done via degree theory, sharp compactness estimates and a topological argument relying on the Poincar\'e-Hopf theorem.