## – Europe/Lisbon

Room P3.10, Mathematics Building — Online

Ederson Moreira dos Santos, ICMC, Universidade de São Paulo em São Carlos

On Hamiltonian systems with critical Sobolev exponents

In this talk I will report some results about lower order perturbations of the critical Lane-Emden system posed on a bounded smooth domain $\Omega \subset \mathbb{R}^N$, with $N \geq3$, inspired by the classical results of Brézis and Nirenberg of 1993. We solve the problem of finding a positive solution for all dimensions $N \geq 4$. For the critical dimension $N=3$ we show a new phenomenon, not observed for scalar problems. Namely, there are parts on the critical hyperbola where solutions exist for all $1$-homogeneous or subcritical superlinear perturbations and parts where there are no solutions for some of those perturbations.