  In this paper, we study the following Schr\"odinger equations with potentials
and general nonlinearities \begin{equation*} \left\{\begin{aligned} & -\Delta
u+V(x)u+\lambda u=|u|^{q-2}u+\beta f(u), \\ & \int |u|^2dx=\Theta,
\end{aligned} \right. \end{equation*} both on $\mathbb{R}^N$ as well as on
domains $r \Omega$ where $\Omega \subset \mathbb{R}^N$ is an open bounded
convex domain and $r>0$ is large. The exponent satisfies $2+\frac{4}{N}\leq
q\leq2^*=\frac{2 N}{N-2}$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies
$L^2$-subcritical or $L^2$-critical growth. This paper generalizes the
conclusion of Bartsch et al. in \cite{TBAQ2023}(2023, arXiv preprint).
Moreover, we consider the Sobolev critical case and $L^2$-critical case of the
above problem.
