# 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.