Normalized ground states for the Sobolev critical Schrödinger equation with at least mass critical growth

Abstract In the present paper, we investigate the existence of ground state solutions to the Sobolev critical nonlinear Schrödinger equation − Δ u + λ u = g u + | u | 2 ∗ − 2 u in R N , ∫ R N | u | 2 d x = m 2 , where N ⩾ 3 , m > 0, 2 ∗ := 2 N N − 2 , λ is an unknown parameter that will appear as a Lagrange multiplier, g is a mass critical or supercritical but Sobolev subcritical nonlinearity. With the aid of the minimization of the energy functional over a linear combination of the Nehari and Pohozaev constraints intersected with the product of the closed balls in L 2 ( R N ) of radii m and the profile decomposition, we obtain a couple of the normalized ground state solution to ( P m ) that is independent of the sign of the Lagrange multiplier. This result complements and extends the paper by Bieganowski and Mederski (2021 J. Funct. Anal. 280 108989) concerning the above problem from the Sobolev subcritical setting to the Sobolev critical framework. We also answer an open problem that was proposed by Jeanjean and Lu (2020 Calc. Var. PDE 59 174). Furthermore, the asymptotic behavior of the ground state energy map is also studied.