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We study existence and properties of ground states for the nonlinearSchr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambdau + \mu |u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,}\] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2, \] in the\emph{Sobolev critical case}. For a $L^2$-subcritical, $L^2$-critical, of$L^2$-supercritical perturbation $\mu |u|^{q-2} u$ we prove severalexistence/non-existence and stability/instability results.  This study can be considered as a counterpart of the Brezis-Nirenberg problemin the context of normalized solutions, and seems to be the first contributionregarding existence of normalized ground states for the Sobolev critical NLSEin the whole space $\mathbb{R}^N$.

Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case