Vectorial solutions for a class of Hartree-Fock type systems with the double coupled feature

In this paper, we study the Hartree-Fock type system as follows: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda \phi _{u,v}u=\left\vert u\right\vert ^{p-2}u+\beta \left\vert v\right\vert ^{\frac{p}{2}}\left\vert u\right\vert ^{\frac{p}{2} -2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta v+v+\lambda \phi _{u,v}v=\left\vert v\right\vert ^{p-2}v+\beta \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 _{u,v}(x)= \int_{\mathbb{R}^{3}}\frac{u^{2}(y)+v^{2}\left( y\right) }{|x-y|}dy, $ the parameters $\lambda,\beta > 0$ and $2 < p < 4$. Such a system is viewed as an approximation of the Coulomb system with two particles appeared in quantum mechanics, taking into account the Pauli principle. Its characteristic feature lies on the presence of the double coupled terms. When $2 < p < 3,$ we establish the existence and multiplicity of nontrivial radial solutions, including vectorial ones, in the radial space $ H_{r}:=H_{rad}^{1}(\mathbb{R}^{3})\times H_{rad}^{1}(\mathbb{R}^{3}) $ by describing the internal relationship between the coupling constants $\lambda $ and $\beta.$ When $2 < p < 4,$ we study the existence of vectorial solutions in the non-radial space $H:=H^{1}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3})$ by developing a novel constraint method, together with some new analysis techniques. In particular, when $3\leq p < 4,$ a vectorial ground state solution is found in $H$, which is innovative as it was not discussed at all in any previous results. Our study can be regarded as an entire supplement in d'Avenia et al. [J. Differential Equations 335 (2022) 580–614].