We study the Brézis-Nirenberg problem under the \(L^2\) constraint \(\int _{\Omega }|u|^2dx = c\) where c is a prescribed positive number. We show that, for any \(j \in {\mathbb {N}}\) , there exists \(c_j > 0\) such that if \(c \in (0,c_j)\) , this problem has at least j sign-changing normalized solutions. The main tools are a new kind of linking contained in an open set of a Hilbert-Riemannian manifold below a level set and the estimates of energies of the sign-changing critical points. Further, these tools do not depend on that the corresponding functional is even, and can be extended to the cases for non-even functionals.
