A global branch approach to normalized solutions for the Schrödinger equation

We study the existence, non-existence and multiplicity of prescribed mass positive solutions to a Schr & ouml;dinger equation of the form -Delta u+lambda u=g(u),u is an element of H-1(R-N),N >= 1. Our approach permits to handle in a unified way nonlinearities g(s) which are either mass subcritical, mass critical or mass supercritical. Among its main ingredients is the study of the asymptotic behaviors of the positive solutions as lambda -> 0+ or lambda ->+infinity and the existence of an unbounded continuum of solutions in (0,+infinity)xH1(R-N).