| Abstract: |
| We consider in this talk the celebrated Brezis-Nirenberg equation in the non-coercive case $\lambda > \Lambda_1$, where $\Lambda_1$ is the first eigenvalue of the Laplacian on a bounded open set of $\mathbb{R}^n$. We prove in dimension 3 the existence of least-energy sign-changing solutions under a positive mass condition. This is the first general existence result for the Brezis-Nirenberg problem in dimension 3 in the non-coercive case. We introduce for this a new non-smooth variational problem, inspired from eigenvalue-optimisation problems in conformal geometry and we show that its minimisers, when they exist, provide least-energy solutions of the Brezis-Nirenberg problem. This is joint work with H. Cheikh Ali (Universite de Lille). |
|