Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity

We investigate normalized solutions for a class of nonlinear Schr\"{o}dinger (NLS) equations with potential $V$ and inhomogeneous nonlinearity $g(|u|)u=|u|^{q-2}u+\beta |u|^{p-2}u$ on a bounded domain $\Omega$. Firstly, when $2+\frac{4}{N}<q<p\leq2^*:=\frac{2N}{N-2}$ and $\beta=-1$, under an explicit smallness assumption on $V$, we prove the existence of a global minimum solution and a high-energy solution if the mass is large enough. For this case we do not require that $\Omega$ is star-shaped, which partly solves an open problem by Bartsch et al. [Math. Ann. 390 (2024) 4813--4859]. Moreover, we find that the global minimizer also exists although the nonlinearity is $L^2$-supercritical. Secondly, when $2<q<2+\frac{4}{N}<p=2^*$ and $\beta=1$, under the smallness and some extra assumptions on $V$, we prove the existence of a ground state and a high-energy solution if $\Omega$ is star-shaped and the mass is small enough. It seems to be new in the study of normalized ground state in the context of the Br\'{e}zis-Nirenberg problem, even for the autonomous case of $V(x)\equiv0$.