A natural constraint approach to normalized solutions of nonlinear SchrA¶dinger equations and systems

Abstract

The paper deals with the existence of normalized solutions to the system −Δu−λ1u=Î1⁄41u3+Î2uv2in R3−Δv−λ2v=Î1⁄42v3+Î2u2vin R3∫R3u2=a12and∫R3v2=a22 for any Î1⁄41,Î1⁄42,a1,a2>0 and Î2<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 Î2→−∞, 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.