In this paper, we are concerned with the nonlinear Schrödinger equation \(\begin{aligned} -\Delta u+V(x)u+\lambda u=g(u)\text { in }{\mathbb {R}}^{N}\text {, }\lambda \in {\mathbb {R}}, \end{aligned}\) with prescribed \(L^{2}\) -norm \(\int _{{\mathbb {R}}^{N}}u^{2}dx=\rho ^{2}\) and \( \lim _{|x|\rightarrow +\infty }V(x)=:V_{\infty }\le +\infty \) under general assumptions on g which allows at least mass critical growth. For the case of \(V_{\infty }<\infty \) , including singular potential, the sufficient conditions are given for the existence of a ground state solution by developing the minimization methods with constraints proposed in Bieganowski and Mederski (J Funct Anal 280(11):108989, 2021) and a delicate analysis of estimates on the least energy comparing with the limiting functional. While for the trapping case \(V_{\infty }=\infty \) , the existence of a ground state solution as well as a second solution of mountain pass type is established.
