Existence and symmetry breaking of vectorial ground states for Hartree-Fock type systems with potentials
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909279411240960 |
|---|---|
| author | Sun, Juntao Wu, Tsung-fang |
| author_facet | Sun, Juntao Wu, Tsung-fang |
| contents | In this paper we study the Hartree-Fock type system as follows: \begin{equation*} \left\{ \begin{array}{ll} -Δu+V\left( x\right) u+ρ\left( x\right) ϕ_{ρ,\left(u,v\right) }u=\left\vert u\right\vert ^{p-2}u+β\left\vert v\right\vert^{\frac{p}{2}}\left\vert u\right\vert ^{\frac{p}{2}-2}u & \text{ in }\mathbb{R}^{3}, \\ -Δv+V\left( x\right) v+ρ\left( x\right) ϕ_{ρ,\left( u,v\right) }v=\left\vert v\right\vert ^{p-2}v+β\left\vert u\right\vert ^{\frac{p}{2}}\left\vert v\right\vert ^{\frac{p}{2}-2}v & \text{ in }\mathbb{R}^{3}, \end{array} \right. \end{equation*} where $ϕ_{ρ,\left( u,v\right) }=\int_{\mathbb{R}^{3}}\frac{ρ\left( y\right) \left( u^{2}(y)+v^{2}\left( y\right) \right) }{|x-y|}dy,$ the potentials $V(x),ρ(x)$ are positive continuous functions in $\mathbb{R}^{3},$ the parameter $β\in \mathbb{R}$ and $2<p<4$. Such system is viewed as an approximation of the Coulomb system with two particles appeared in quantum mechanics, whose main characteristic is the presence of the double coupled terms. When $2<p<3,$ under suitable assumptions on potentials, we shed some light on the behavior of the corresponding energy functional on $H^{1}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3}),$ and prove the existence of a global minimizer with negative energy. When $3\leq p<4,$ we find vectorial ground states by developing a new analytic method and exploring the conditions on potentials. Finally, we study the phenomenon of symmetry breaking of ground states when $2<p<3.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01786 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and symmetry breaking of vectorial ground states for Hartree-Fock type systems with potentials Sun, Juntao Wu, Tsung-fang Analysis of PDEs In this paper we study the Hartree-Fock type system as follows: \begin{equation*} \left\{ \begin{array}{ll} -Δu+V\left( x\right) u+ρ\left( x\right) ϕ_{ρ,\left(u,v\right) }u=\left\vert u\right\vert ^{p-2}u+β\left\vert v\right\vert^{\frac{p}{2}}\left\vert u\right\vert ^{\frac{p}{2}-2}u & \text{ in }\mathbb{R}^{3}, \\ -Δv+V\left( x\right) v+ρ\left( x\right) ϕ_{ρ,\left( u,v\right) }v=\left\vert v\right\vert ^{p-2}v+β\left\vert u\right\vert ^{\frac{p}{2}}\left\vert v\right\vert ^{\frac{p}{2}-2}v & \text{ in }\mathbb{R}^{3}, \end{array} \right. \end{equation*} where $ϕ_{ρ,\left( u,v\right) }=\int_{\mathbb{R}^{3}}\frac{ρ\left( y\right) \left( u^{2}(y)+v^{2}\left( y\right) \right) }{|x-y|}dy,$ the potentials $V(x),ρ(x)$ are positive continuous functions in $\mathbb{R}^{3},$ the parameter $β\in \mathbb{R}$ and $2<p<4$. Such system is viewed as an approximation of the Coulomb system with two particles appeared in quantum mechanics, whose main characteristic is the presence of the double coupled terms. When $2<p<3,$ under suitable assumptions on potentials, we shed some light on the behavior of the corresponding energy functional on $H^{1}(\mathbb{R}^{3})\times H^{1}(\mathbb{R}^{3}),$ and prove the existence of a global minimizer with negative energy. When $3\leq p<4,$ we find vectorial ground states by developing a new analytic method and exploring the conditions on potentials. Finally, we study the phenomenon of symmetry breaking of ground states when $2<p<3.$ |
| title | Existence and symmetry breaking of vectorial ground states for Hartree-Fock type systems with potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.01786 |