In this paper, we are concerned with the following Schrödinger-Poisson systems \({\left\{ \begin{array}{ll} -\Delta u +\alpha \phi u= \lambda u+\mu |u|^{q-2}u+|u|^{p-2}u,& ~~ \text{ in }~\Omega ,\\ -\Delta \phi =u^2,& ~~ \text{ in }~\Omega ,\\ u=\phi =0,& ~\text{ on }~\partial \Omega ,\\ \end{array}\right. } \) with prescribed \(L^{2}\) -norm mass \(\begin{aligned} \int _{\Omega } |u|^2dx=c^2, \end{aligned}\) where \(2< q<p\le 6\) , \(\alpha \in \mathbb {R}\) is a parameter, c is a prescribed value, \(\lambda \in \mathbb {R}\) is a Lagrange multiplier, \(\Omega \subset \mathbb {R}^3\) is a smooth bounded domain and \(p=6\) is the Sobolev critical exponent. We first prove that the problem has a positive normalized solution, which is a local minimizer. Next, under the assumption that \(\Omega \) is star-shaped, we show the existence of a second normalized solution for \(\alpha <0\) and \(4\le p< 6\) by using Jeanjean’s theory, Pohozaev identity and Mountain pass theorem. Additionally, we give asymptotic behavior of the local minimizer as \(c\rightarrow 0\) .
