The paper deals with the existence and orbital stability for standing waves of nonlinear Schrödinger equations \[ iu_t+\Delta u+f(u)=0 \] where \((t,x)\in \mathbb{R}\times \mathbb{R}^N\) and \(N\geq 1\). The corresponding minimizing problem on existence and the non-existence of global minimizers of \[ E_{\alpha}=\inf _{u\in M_{\alpha}}I[u], \] where \[ M_{\alpha}=\{ u\in H^1(\mathbb{R}^N); \|u\|^2_{L^2(\mathbb{R}^N)}=\alpha \}, \]
\[ I[u]=\frac{1}{2}\int_{R^N}|\nabla u|^2 dx-\int_{\mathbb{R}^N}F(|u|) dx \] and \[ F(s)=\int_0^s f(\tau)d \tau \] is studied. It is proved that there exists \(\alpha_0\geq 0\) such that there exists a global minimizer if \(\alpha >\alpha_0\) and there exists no global minimizer if \(\alpha <\alpha_0\). Conditions for \(\alpha_0=0\) or \(\alpha_0>0\) are derived and the existence results with respect to \(E_{\alpha_0}\) are obtained also.