In this paper, we are concerned with the nonlinear Schrödinger equation

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.

在本文中，我们关注的是非线性薛定谔方程

具有规定的\(L^{2}\) -范数\(\int _{{\mathbb {R}}^{N}}u^{2}dx=\rho ^{2}\)和\( \ lim _{|x|\rightarrow +\infty }V(x)=:V_{\infty }\le +\infty \)在g的一般假设下，至少允许质量临界增长。对于\(V_{\infty }<\infty \)的情况，包括奇异势，通过开发 Bieganowski 和 Mederski (J Funct Anal) 中提出的约束最小化方法，给出了基态解存在的充分条件280(11):108989, 2021）并对最小能量与极限泛函的估计进行了细致的分析。而对于捕获情况\(V_{\infty }=\infty \)，则建立了基态解以及山口类型的第二解的存在性。