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

Andrea Malchiodi (1) , Luca Martinazzi (2) , Pierre-Damien Thizy (3)

1 SNS - Scuola Normale Superiore di Pisa
2 UNIROMA - Università degli Studi di Roma "La Sapienza" = Sapienza University [Rome]
3 EDPA - Équations aux dérivées partielles, analyse

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.

Domaines: Equations aux dérivées partielles [math.AP]

Andrea Malchiodi, Luca Martinazzi, Pierre-Damien Thizy. Critical points of arbitrary energy for the Trudinger-Moser functional in planar domains. Advances in Mathematics, inPress. ⟨hal-03908236⟩