Normalized ground states for Sobolev critical nonlinear Schrödinger equation in the L² -supercritical case

In this paper, we study the existence of the normalized ground state solutions to Sobolev critical nonlinear Schrödinger equation:arrayll \ arrayll -Δ u+λ u = f(u)+|u|2^*-2u, &in\;RN,\\ ∫RN|u|²dx = m², array . array(Pₘ)where N≥ 3, 2^*: = 2N/N-2, $ m>0 $, λ is unknown and will appear as a Lagrange multiplier, f is a mass supercritical and Sobolev subcritical nonlinearity. Using Pohozaev manifold and the concentration-compactness principle, we obtain a couple of the normalized solution to (Pₘ). The main contribution is related to the fact that we extend the results of L. Jeanjean, S. Lu published in 2020 on Calc. Var. [21] concerning the above problem from Sobolev subcritical setting to Sobolev critical setting, and our results answer an open problem raised by N. Soave published in 2020 on J. Funct. Anal. [37].