We consider in this talk the celebrated Brézis-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 $R^n$. We prove in dimension 3, and on a general bounded set, the existence of least-energy sign-changing solutions of very small energy when $\lambda$ belongs to a  left-neighbourhood of any eigenvalue, which is explicitly characterized by a positive mass condition. This is, in particular, the first general existence result for the Brézis-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. Precisely, we consider the principal eigenvalue of $- \Delta - \lambda$ over an appropriate weighted $L^2$ space and minimize its value over the set of all admissible weight. We develop an ad hoc variational theory for this problem and show that its minimisers, when they exist, provide least-energy solutions of the Brézis-Nirenberg problem. Our framework applies in every dimension and when $n \ge 4$ we also show that the energy function of the Brézis-Nirenberg problem is discontinuous exactly at the spectrum of $-\Delta$. This is joint work with H. Cheikh Ali (Université de Lille) and is based on arXiv:2509.19145.