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

Abstract

We investigate normalized solutions for a class of nonlinear Schrödinger (NLS) equations with potential $V$ and inhomogeneous nonlinearity $g(|u|)u=|u{|}^{q-2}u+{\hat{I}}^2|u{|}^{p-2}u$ on a bounded domain $\hat{I}\text{c◯}$. Firstly, when $2+\frac{4}{N}<q<p\le 2^{\ast }:=\frac{2N}{N-2}$ and ${\hat{I}}^2=-1$, 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 $\hat{I}\text{c◯}$ 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 $2<q<2+\frac{4}{N}<p=2^{\ast }$ and ${\hat{I}}^2=1$, under the smallness and some extra assumptions on $V$, we prove the existence of a ground state and a high-energy solution if $\hat{I}\text{c◯}$ 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 Brézis-Nirenberg problem, even for the autonomous case of $V(x)\equiv 0$.