NORMALIZED GROUND STATES FOR SOBOLEV CRITICAL NONLINEAR SCHRODINGER EQUATION IN THE L2-SUPERCRITICAL CASE

In this paper, we study the existence of the normalized ground state solutions to Sobolev critical nonlinear Schrodinger equation: { -triangle u+lambda u=f(u) +|u|(2 & lowast;)-2u,inR(N),(P-m)integral R-N|u|(2)dx=m(2),whereN >= 3, 2 & lowast;:=2N/N-2,m >0,lambda is unknown and will appear as a Lagrange multiplier, fis a mass supercritical and Sobolev subcritical nonlinearity. Using Pohozaev manifold and the concentration-compactness principle, we obtain acouple 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 settingto Sobolev critical setting, and our results answer an open problem raised by N. Soave published in 2020 on J. Funct. Anal. [37]

Automated summary

In this paper, we investigate the existence of normalized ground state solutions to the Sobolev critical nonlinear Schrödinger equation. By using the Pohozaev manifold and the concentration-compactness principle, we obtain a couple of normalized solutions to the equation. Our main contribution is extending the previous results and answering an open problem raised by N. Soave.