A note on deformation argument for $L^2$ normalized solutions of nonlinear Schrödinger equations and systems

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 constrained functional on $S_m=\{ u; \, \int_{\mathbb R^N} |u^2 | =m\}$ or $S_{m_1}\times S_{m_2}$. 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.