A natural constraint approach to normalized solutions of nonlinear Schr\"odinger equations and systems

TL;DR

This paper establishes the existence of normalized solutions for coupled nonlinear Schrödinger equations with prescribed parameters using a natural constraint method, simplifying previous proofs and applicable to scalar cases.

Contribution

It introduces a natural constraint approach to find normalized solutions, providing new and simplified proofs for coupled and scalar nonlinear Schrödinger equations.

Findings

Existence of solutions for prescribed parameters in coupled NLS equations.

Method applicable to scalar NLS with normalization constraints.

Simplified proofs compared to existing literature.

Abstract

We prove existence of normalized solutions to \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 & \text{in $\mathbb{R}^3$} -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v & \text{in $\mathbb{R}^3$}\int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a_2^2 \end{cases} \] for any $\mu_1,\mu_2,a_1,a_2>0$ and $\beta<0$ prescribed. The approach is based upon the introduction of a natural constraint associated to the problem. Our method can be adapted to the scalar NLS equation with normalization constraint, and leads to alternative and simplified proofs to some results already available in the literature.