Qualitative analysis for Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent

Abstract

In this paper, we develop an exhaustive analysis on standing waves with prescribed mass for the coupled Hartree-Fock system as following, which is introduced by Hartree in the 1920's and developed by Fock for describing large systems of identical fermions,$\{\begin{array}{cc}& -\Delta u+{\lambda }_{1}u=\left(\right|x{|}^{-\alpha }\star \left|u{|}^2\right)u+{\mu }_1\left(\right|x{|}^{-\gamma }\star \left|u{|}^2\right)u+\beta \left(\right|x{|}^{-\gamma }\star \left|v{|}^2\right)u\\ & -\Delta v+{\lambda }_{2}v=\left(\right|x{|}^{-\alpha }\star \left|v{|}^2\right)v+{\mu }_2\left(\right|x{|}^{-\gamma }\star \left|v{|}^2\right)v+\beta \left(\right|x{|}^{-\gamma }\star \left|u{|}^2\right)v\end{array}\text{in }{R}^5$ under mass constraint conditions$\underset{R^5}{\int }|u{|}^2=c^2,\underset{R^5}{\int }|v{|}^2=d^2,$ where ${\mu }_i>0(i=1,2),\beta >0$, $0<\alpha <\gamma \le 4$, and the frequency ${\lambda }_i\in R(i=1,2)$ are a part of unknown and appear as Lagrange multipliers. Firstly, when $0<\alpha <\gamma =2$ or $\alpha =2<\gamma <4$, we establish an existence result by using the constrained minimization method. Then turning to the case of $0<\alpha <2<\gamma <4$ or $0<\alpha <2,\gamma =4$, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise mass collapse behaviors of the ground state solutions for $0<\alpha <2<\gamma <4$ and $0<\alpha <2,\gamma =4$ when the masses of two components vanish at the same rate. We also give a precise limit profiles of the mountain pass solutions when $0<\alpha <2,\gamma =4$ as $(c,d)\to (0,0)$. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behaviors of the normalized solutions to Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent. Finally, when $2\le \alpha <4$ and $\gamma =4$, we prove an existence result of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise limit profiles of the ground states are obtained when the masses of two components vanish. These results are a continuation of the works [5] and [14] concerning normalized solutions for Hartree equations to Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent.