# 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.