For N≥6, the normalized Sobolev-critical NLS with potential has a positive mountain-pass solution; for N≥3, a negative-energy local minimizer exists under weaker conditions than in the cited preprint.
Ghoussoub, Duality and perturbation methods in critical point theory, Cambridge Uni- versity Press, Cambridge, 1993
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Normalized solutions for the NLS equation with potential in higher dimension: the purely Sobolev critical case
For N≥6, the normalized Sobolev-critical NLS with potential has a positive mountain-pass solution; for N≥3, a negative-energy local minimizer exists under weaker conditions than in the cited preprint.