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

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities−Δu=λu+μ|u|q−2u+|u|2⁎−2uin RN, N≥3, having prescribed mass∫RN|u|2=a2, in the Sobolev critical case. For a L2-subcritical, L2-critical, of L2-supercritical perturbation μ|u|q−2u we prove several existence/non-existence and stability/instability results. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space RN.