Existence of normalized solutions of a Hartree-Fock system with mass subcritical growth

Abstract

In this paper, we are concerned with normalized solutions in $H_{r}^{1}(\mathbb{R}^{3}) \times H_{r}^{1}(\mathbb{R}^{3})$ for Hartree-Fock type systems with the form \be\lab{ Hartree-Fock} \left\{ \begin{array}{ll} -\Delta u +\alpha \phi _{u,v} u=\lambda _{1} u+\left | u \right | ^{2q-2} u+\beta \left | v \right | ^{q} \left | u \right | ^{q-2} u , \\ -\Delta v +\alpha \phi _{u,v} v=\lambda _{2} v+\left | v\right | ^{2q-2} v+\beta \left | u \right | ^{q} \left | v \right | ^{q-2} v , \\ \int_{\mathbb{R}^{3}}\left | u \right | ^{2} {\rm d}x=a_{1} , \quad \int_{\mathbb{R}^{3}}\left | v \right | ^{2} {\rm d}x=a_{2} , \nonumber\\ \end{array} where $$ \phi_{u, v}\left(x\right):=\int_{\mathbb{R}^{3}} \frac{u^{2}(y)+v^{2}(y)}{|x-y|} {\rm d}y \in D^{1,2}\left(\mathbb{R}^{3}\right). $$ Here $\alpha,\beta>0, a_1,a_2>0$ and $1<q<\frac{5}{3}$. By seeking the constrained global minimizers of the corresponding functional, we prove that the existence of normalized solutions to the system above for any $a_1,a_2>0$ when $1<q<\frac{4}{3}$ and for $a_1,a_2>0$ small when $\frac{4}{3}\le q < \frac{3}{2}$. The nonexistence of normalized solutions is also considered for $\frac{3}{2}\le q < \frac{5}{3}$. Also, the orbital stability of standing waves is obtained under local well-posedness assumptions of the evolution problem.