A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems

Abstract

We prove existence of normalized solutions to \[ \begin{cases} -Δu - λ_1 u = μ_1 u^3+ βu v^2 & \text{in ${\mathbb{R}}^3$} -Δv- λ_2 v = μ_2 v^3 +βu^2 v & \text{in ${\mathbb{R}}^3$}\int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a_2^2 \end{cases} \] 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.

Final version, accepted for publication on J. Functional Analysis