A note on deformation argument for $L^2$ constraint problem

Abstract

We study the existence of L 2 normalized solutions for nonlinear Schrödinger equations and systems. Under new Palais-Smale type conditions we develop new deformation arguments for the constraint functional onAs applications, we give other proofs to the results of [20,6,7]. As to the results of [20,6], our deformation result enables us to apply the genus theory directly to the corresponding functional to obtain infinitely many solutions. As to the result [7], via our deformation result we can show the existence of vector solution without using constraint related to the Pohozaev identity.