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 Schrödinger equation of the form$-\Delta u+\lambda u=g\left(u\right),u\in H^1\left(R^N\right),N\ge 1.$ Our approach permits to handle in a unified way nonlinearities $g\left(s\right)$ 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 $\lambda \to 0^{+}$ or $\lambda \to +\infty$ and the existence of an unbounded continuum of solutions in $(0,+\infty )\times H^1\left(R^N\right)$.