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

Abstract

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