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

In this paper, we are concerned with normalized solutions in $H_r^1({\mathbb{R}}^3)\times H_r^1({\mathbb{R}}^3)$ for Hartree-Fock type systems with the form \be\lab{ Hartree-Fock} \left{ \begin{array}{ll} -Δu +αφ_{u,v} u=λ_{1} u+\left | u \right | ^{2q-2} u+β\left | v \right | ^{q} \left | u \right | ^{q-2} u , \ -Δv +αφ_{u,v} v=λ_{2} v+\left | v\right | ^{2q-2} v+β\left | u \right | ^{q} \left | v \right | ^{q-2} v , \ \int_{$\mathbb{R}$^{3}}\left | u \right | ^{2} {\rm d}x=a_{1} , \quad \int_{$\mathbb{R}$^{3}}\left | v \right | ^{2} {\rm d}x=a_{2} , \nonumber\ \end{array} where $${\phi }_{u,v}\left(x\right):={\int }_{{\mathbb{R}}^3}\frac{u^2(y)+v^2(y)}{\mathrm{\mid }x-y\mathrm{\mid }}\mathrm{d}y\in D^{1,2}\left({\mathbb{R}}^3\right)\mathrm{.}$$ Here $\alpha ,\beta >0,a_1,a_2>0$ and $1<q<\frac{5}{3}$. By seeking the constrained global minimizers of the corresponding functional, we prove that the existence of normalized solutions to the system above for any $a_1,a_2>0$ when $1<q<\frac{4}{3}$ and for $a_1,a_2>0$ small when $\frac{4}{3}\le q<\frac{3}{2}$. The nonexistence of normalized solutions is also considered for $\frac{3}{2}\le q<\frac{5}{3}$. Also, the orbital stability of standing waves is obtained under local well-posedness assumptions of the evolution problem.