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

Abstract: In this paper we study the Hartree-Fock type system as follows: \begin{equation*} \left{ \begin{array}{ll} -\Delta u+V\left( x\right) u+\rho \left( x\right) \phi {\rho ,\left(u,v\right) }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\left( x\right) v+\rho \left( x\right) \phi _{\rho ,\left( u,v\right) }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 $\phi _{\rho ,\left( u,v\right) }=\int{\mathbb{R}^{3}}\frac{\rho \left( y\right) \left( u^{2}(y)+v^{2}\left( y\right) \right) }{|x-y|}dy,$ 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 $3\leq p<4,$ 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 $2<p<3.$