Normalised solutions and limit profiles of the defocusing Gross–Pitaevskii–Poisson equation

We consider normalised solutions of the stationary Gross–Pitaevskii–Poisson (GPP) equation with a defocusing local nonlinear term, aligned -Δ u+λ u+|u|²u =(I_α *|u|²)u in R³, ∫ R³u²dx=ρ ², aligned - Δ u + λ u + | u | 2 u = ( I α ∗ | u | 2 ) u in R 3 , ∫ R 3 u 2 d x = ρ 2 , where ρ ²>0 ρ 2 > 0 is the prescribed mass of the solutions, λ ∈ R λ ∈ R is an a-priori unknown Lagrange multiplier, and I_α (x)=A_α |x|3-α I α ( x ) = A α | x | 3 - α is the Riesz potential of order α ∈ (0,3) α ∈ ( 0 , 3 ) . When α =2 α = 2 this problem appears in the models of self–gravitating Bose–Einstein condensates, which were proposed in cosmology and astrophysics to describe Cold Dark Matter and Boson Stars. We establish the existence of branches of normalised solutions to the GPP equation, paying special attention to the shape of the associated mass–energy relation curves and to the limit profiles of solutions at the endpoints of these curves. The behaviour of normalised solutions depends sensitively on whether