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

Given a smooth bounded domain Ω⊂R³, we consider the following nonlinear Schrödinger-Poisson type system {−Δu+ϕu−|u|^p−2u=ωuin λΩ,−Δϕ=u²in λΩ,u>0in λΩ,u=ϕ=0on ∂(λΩ),∫_λΩu²dx=ρ² in the expanding domain λΩ⊂R³,λ>1 and p∈(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→+∞ the solutions found converge to a ground state of the problem in the whole space R³.