Normalized Solutions on large smooth domains to the Schrödinger equation with potential and general nonlinearity: Mass super-critical case

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrödinger equation with general nonlinearity: Mass super-critical case: $$\{\begin{array}{l}-\mathrm{\Delta }u+V(x)u+\lambda u=g(u),\\ \mathrm{\parallel }u{\mathrm{\parallel }}_2^2=\int \mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{d}x=c,\end{array}$$ both on large bounded smooth star-shaped domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ and on ${\mathbb{R}}^N$, where $V(x)$ is the potential and the nonlinearity $g(\cdot )$ considered here are very general and of mass super-critical. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential $V(x)$. In addition, our study can be considered as a complement of Bartsch-Qi-Zou (Math Ann 390, 4813–4859, 2024), which has addressed an open problem raised in Bartsch et al. (Commun Partial Differ Equ 46(9):1729–1756, 2021).