We study the existence of L² normalized solutions for nonlinear Schrödinger equations and systems. Under new Palais-Smale type conditions, we develop new deformation arguments for the constrained functional on Sₘ=\ u; \, ∫RN |u² | =m\ or Sm₁× Sm₂. As applications, we give other proofs to the results of [5,8, 22]. As to the results of [5, 22], our deformation result enables us to apply the genus theory directly to the corresponding functional to obtain infinitely many solutions. As to the result [8], via our deformation result, we can show the existence of vector solution without using constraint related to the Pohozaev identity.
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Ikoma et al. (2019) studied this question.