Small normalised solutions for a Schr\"odinger-Poisson system in expanding domains: multiplicity and asymptotic behaviour

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