Andrea Malchiodi - On critical points of the Moser-Trudinger functional

It is known that in two dimensions Sobolev functions in $W^{1,2}$ satisfy critical embedding properties of exponential type. In 1971 Moser obtained a sharp form of the embedding, controlling the integrability of $F(u) := \int \exp(u^2)$ in terms of the Sobolev norm of $u$. On a closed Riemannian surface, $F(u)$ is unbounded above for $\|u\|_{W^{1,2}} > 4 \pi$.  We are however able to find critical points of $F$ constrained to any sphere  $\{ \|u\|_{W^{1,2}} = \beta \}$, with $\beta > 0$ 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.