Andrea Malchiodi

Scuola Normale Superiore

On critical points of the Moser–Trudinger functional

Since the fundamental work by Trudinger from 1967 it is known that in two dimensions Sobolev functions in \(W^{1,2}\) satisfy embedding properties of exponential type. In 1971 Moser then obtained a sharp form of the embedding, controlling the integrability of \(F(u) := ∫ \exp(u^2)\) in terms of the Sobolev norm of \(u\).

On a closed Riemannian surface, \(F(u)\) is unbounded above for \(\|u\|_{W^{1,2}} > 4π\). We are however able to find critical points of \(F\) constrained to any sphere \(\{ \|u\|_{W^{1,2}} = β \}\), with \(β > 0\) arbitrary. The proof combines min-max theory, a monotonicity argument by Struwe, blow-up analysis and compactness estimates. This is joint work with F. De Marchis, O. Druet, L. Martinazzi and P. D. Thizy.