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

## TL;DR

This paper develops new deformation arguments under Palais-Smale conditions to establish the existence of $L^2$ normalized solutions for nonlinear Schrödinger equations, providing alternative proofs and extending solution existence results.

## Contribution

Introduces new deformation techniques for constraint functionals, enabling direct application of genus theory and proving existence of solutions without Pohozaev constraints.

## Key findings

- Established existence of $L^2$ normalized solutions.
- Provided alternative proofs for previous results.
- Extended solution existence to vector solutions without Pohozaev constraints.

## Abstract

We study the existence of $L^2$ normalized solutions for nonlinear Schr\"odinger equations and systems. Under new Palais-Smale type conditions we develop new deformation arguments for the constraint functional on $S_m=\{ u; \, \int_{\mathbf{R}^N} | u |^2=m\}$ or $S_{m_1} \times S_{m_2}$. As applications, we give other proofs to the results of [\cite[J:20], \cite[BdV:6], \cite[BS1:7]]. As to the results of [\cite[J:20], \cite[BdV: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 [\cite[BS1:7]], via our deformation result we can show the existence of vector solution without using constraint related to the Pohozaev identity.