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

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 Poincaré-Hopf theorem.