Existence and symmetry breaking of vectorial ground states for Hartree-Fock type systems with potentials

In this paper we study the Hartree-Fock type system as follows: \begin{equation*} \left{ \begin{array}{ll} -Δu+V\left( x\right) u+ρ\left( x\right) φ_{ρ,\left(u,v\right) }u=\left\vert u\right\vert ^{p-2}u+β\left\vert v\right\vert^{\frac{p}{2}}\left\vert u\right\vert ^{\frac{p}{2}-2}u & $\text{ in }$$\mathbb{R}$^{3}, \ -Δv+V\left( x\right) v+ρ\left( x\right) φ_{ρ,\left( u,v\right) }v=\left\vert v\right\vert ^{p-2}v+β\left\vert u\right\vert ^{\frac{p}{2}}\left\vert v\right\vert ^{\frac{p}{2}-2}v & $\text{ in }$$\mathbb{R}$^{3}, \end{array} \right. \end{equation*} where ${\phi }_{\rho ,(u,v)}={\int }_{{\mathbb{R}}^3}\frac{\rho \left(y\right)(u^2(y)+v^2\left(y\right))}{\mathrm{\mid }x-y\mathrm{\mid }}dy,$ the potentials $V(x),\rho (x)$ are positive continuous functions in ${\mathbb{R}}^3,$ the parameter $\beta \in \mathbb{R}$ and $2<p<4$. Such system is viewed as an approximation of the Coulomb system with two particles appeared in quantum mechanics, whose main characteristic is the presence of the double coupled terms. When $2<p<3,$ under suitable assumptions on potentials, we shed some light on the behavior of the corresponding energy functional on $H^1({\mathbb{R}}^3)\times H^1({\mathbb{R}}^3),$ and prove the existence of a global minimizer with negative energy. When $3\le p<4,$ we find vectorial ground states by developing a new analytic method and exploring the conditions on potentials. Finally, we study the phenomenon of symmetry breaking of ground states when $2<p<3.$