Existence of normalized solutions of a Hartree-Fock system with mass subcritical growth

In this paper, we are concerned with normalized solutions in Hᵣ¹(R³) × Hᵣ¹(R³) for Hartree-Fock type systems with the form { Hartree-Fock} \{ {array}{ll} -Δ u +α φ u,v u=λ ₁ u+ | u | 2q-2 u+β | v | q | u | q-2 u , \\ -Δ v +α φ u,v v=λ ₂ v+ | v | 2q-2 v+β | u | q | v | q-2 v , \\ ∫_{R³} | u | ² { d}x=a₁ , ∫_{R³} | v | ² { d}x=a₂ , \\ {array} where φu, v(x):=∫_R³ u²(y)+v²(y)|x-y| dy ∈ D1,2(R³). Here α,β>0, a₁,a₂>0 and 1<q<5/3. By seeking the constrained global minimizers of the corresponding functional, we prove that the existence of normalized solutions to the system above for any a₁,a₂>0 when 1<q<4/3 and for a₁,a₂>0 small when 4/3≤ q < 3/2. The nonexistence of normalized solutions is also considered for 3/2≤ q < 5/3. Also, the orbital stability of standing waves is obtained under local well-posedness assumptions of the evolution problem.