Existence of normalized solutions of a Hartree-Fock system with mass subcritical growth
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2024
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| author | Jin, Hua Chang, Yanyun Squassina, Marco Zhang, Jianjun |
| author_facet | Jin, Hua Chang, Yanyun Squassina, Marco Zhang, Jianjun |
| contents | 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} -Δu +αϕ_{u,v} u=λ_{1} u+\left | u \right | ^{2q-2} u+β\left | v \right | ^{q} \left | u \right | ^{q-2} u , \\ -Δv +αϕ_{u,v} v=λ_{2} v+\left | v\right | ^{2q-2} v+β\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 $$ ϕ_{u, v}\left(x\right):=\int_{\mathbb{R}^{3}} \frac{u^{2}(y)+v^{2}(y)}{|x-y|} {\rm d}y \in D^{1,2}\left(\mathbb{R}^{3}\right). $$ Here $α,β>0, a_1,a_2>0$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_01036 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of normalized solutions of a Hartree-Fock system with mass subcritical growth Jin, Hua Chang, Yanyun Squassina, Marco Zhang, Jianjun Analysis of PDEs 35J50, 35R09, 37K45 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} -Δu +αϕ_{u,v} u=λ_{1} u+\left | u \right | ^{2q-2} u+β\left | v \right | ^{q} \left | u \right | ^{q-2} u , \\ -Δv +αϕ_{u,v} v=λ_{2} v+\left | v\right | ^{2q-2} v+β\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 $$ ϕ_{u, v}\left(x\right):=\int_{\mathbb{R}^{3}} \frac{u^{2}(y)+v^{2}(y)}{|x-y|} {\rm d}y \in D^{1,2}\left(\mathbb{R}^{3}\right). $$ Here $α,β>0, a_1,a_2>0$ 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. |
| title | Existence of normalized solutions of a Hartree-Fock system with mass subcritical growth |
| topic | Analysis of PDEs 35J50, 35R09, 37K45 |
| url | https://arxiv.org/abs/2405.01036 |