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

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