Existence and behavior of minimizers for a class of Hartree–Fock type systems

Abstract

In this paper, we investigate the Hartree–Fock type system: $\left(\begin{array}{c}-\Delta u+\lambda u+\mu {\varphi }_{u,v}u={\left(u\right)}^{q-2}u+\rho {\left(v\right)}^{\frac{q}{2}}{\left(u\right)}^{\frac{q}{2}-2}u,\\ -\Delta v+\lambda v+\mu {\varphi }_{u,v}v={\left(v\right)}^{q-2}v+\rho {\left(u\right)}^{\frac{q}{2}}{\left(v\right)}^{\frac{q}{2}-2}v,\end{array}\right)$where ${\varphi }_{u,v}\left(x\right)={\int }_{R^3}\frac{u^2\left(y\right)+v^2\left(y\right)}{|x-y|}dy,$ the parameters $\mu ,\rho >0$ and $q\in (2,3)$. Such a system is regarded as an approximation of the Coulomb system of two particles that occurs in quantum mechanics. Due to the existence of the nonlocal term ${\varphi }_{u,v}$, we find that in $R^3$, the energy of the minimizer is bounded in the radial case but not in the non-radial case. To further investigate this phenomenon, without loss of generality, we consider the problem in a ball $B_R\left(0\right)\subset R^3$. We study the behavior of minimizers in the radial and non-radial cases which depends on $q,\mu ,\rho ,R$, and obtain three positive vectorial solutions. Our study can be seen as an extension of [1], [2].