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

The paper deals with the existence of normalized solutions to the system −Δu−λ1u=μ1u3+βuv2in R3−Δv−λ2v=μ2v3+βu2vin R3∫R3u2=a12and∫R3v2=a22 for any μ1,μ2,a1,a2>0 and β<0 prescribed. We present a new approach that is based on the introduction of a natural constraint associated to the problem. We also show that, as β→−∞, phase separation occurs for the solutions that we find. Our method can be adapted to scalar nonlinear Schrödinger equations with normalization constraint, and leads to alternative and simplified proofs to some results already available in the literature.