Standing waves with prescribed mass for the coupled Hartree–Fock system with partial confinement

Abstract

In this paper, we focus on the standing waves with prescribed mass for the coupled Hartree–Fock system, which is the basic quantum chemistry model of small number of electrons interacting with static nuclei. This leads to study the normalized solutions to the following nonlocal elliptic system $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+(x_{1}^{2}+x_{2}^{2}+\cdot \cdot \cdot +x_{N-1}^{2})u=\lambda _{1}u+\mu _{1}\phi _{u}|u|^{p-2}u+\beta \phi _{v}|u|^{p-2}u, &{}\quad x\in \mathbb {R}^N,\\ \displaystyle -\Delta v+(x_{1}^{2}+x_{2}^{2}+\cdot \cdot \cdot +x_{N-1}^{2})v=\lambda _{2}v+\mu _{2}\phi _{v}|v|^{p-2}v+\beta \phi _{u}|v|^{p-2}v, &{}\quad x\in \mathbb {R}^N, \end{array}\right. } \end{aligned}$$ where $$N\ge 3$$ , $$\mu _{i}>0$$ (i=1,2), $$\beta >0$$ and the frequencies $$\lambda _{1}$$ , $$\lambda _{2}$$ are unknown and appear as Lagrange multipliers. For this purpose, we firstly give the existence of normalized solutions to the related single equation, which extends the main results in Bellazzini et al. (Commun Math Phys 1:229–251, 2017) to the nonlocal case. Then, combining constrained minimization method and coupled rearrangement technique, we prove the existence of normalized solutions to the above nonlocal elliptic system together with some qualitative properties. In addition, the stability of the corresponding standing waves for the related time-dependent coupled Hartree–Fock system is discussed. These results can be viewed as an extension of the main results in Wang et al. (Discrete Contin Dyn Syst 56, 2019) from mass-subcritical case $$\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}$$ , to mass-supercritical setting $$\frac{N+\alpha +2}{N}<p<\min \{4,\frac{N+\alpha }{N-2}\}$$ .