Small normalised solutions for a Schrödinger-Poisson system in expanding domains: Multiplicity and asymptotic behaviour

Given a smooth bounded domain $\Omega \subset R^3$, we consider the following nonlinear Schrödinger-Poisson type system$\{\begin{array}{cc}-\Delta u+\varphi u-|u{|}^{p-2}u=\omega u& \text{in }\lambda \Omega ,\\ -\Delta \varphi =u^2& \text{in }\lambda \Omega ,\\ u>0& \text{in }\lambda \Omega ,\\ u=\varphi =0& \text{on }\partial \left(\lambda \Omega \right),\\ {\int }_{\lambda \Omega }{u}^2\text{d}x={\rho }^2\end{array}$ in the expanding domain $\lambda \Omega \subset R^3,\lambda >1$ and $p\in (2,3)$, in the unknowns $(u,\varphi ,\omega )$. We show that, for arbitrary large values of the expanding parameter λ and arbitrary small values of the mass $\rho >0$, the number of solutions is at least the Ljusternick-Schnirelmann category of λΩ. Moreover we show that as $\lambda \to +\infty$ the solutions found converge to a ground state of the problem in the whole space $R^3$.