Normalized solutions for a system of coupled cubic Schrödinger equations on R3

Abstract

We consider the system of coupled elliptic equations−Δu−λ1u=μ1u3+βuv2−Δv−λ2v=μ2v3+βu2vin R3, and study the existence of positive solutions satisfying the additional condition∫R3u2=a12and∫R3v2=a22. Assuming that a1,a2,μ1,μ2 are positive fixed quantities, we prove existence results for different ranges of the coupling parameter β>0. The extension to systems with an arbitrary number of components is discussed, as well as the orbital stability of the corresponding standing waves for the related Schrödinger systems