Existence of normalized ground states for the Sobolev critical Schrödinger equation with combined nonlinearities

We study the Sobolev critical Schrödinger equation with combined power nonlinearities - Δ u = λ u + | u | 2 N N - 2 - 2 u + μ | u | q - 2 u , x ∈ R N having prescribed mass ∫ R N | u | 2 d x = a 2 . For a L 2 -critical or L 2 -supercritical perturbation μ | u | q - 2 u , we prove existence of normalized ground states, by introducing the Sobolev subcritical approximation method to mass constrained problem. Our result settles a question raised by N. Soave [ 22 ]. Meanwhile, the Sobolev subcritical problem is treated again by using the Pohožaev constraint, Schwartz symmetrization rearrangements and various scaling transformations.