Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
In this article, we are concerned with the existence, non-existence, and blow-up behavior of normalized ground state solutions for the mass critical Hartree-Fock type Schrödinger equation with rotation i∂tu=−Δu+2V(x)u+2ΩLzu−λu−bu∫RN∣u(y)∣2∣x−y∣2dy,(t,x)∈R×RN,u(0,x)=u0(x),\left\{\begin{array}{l}i{\pa...
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De Gruyter
2025-06-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2025-0085 |
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| author | Tu Yuanyuan Wang Jun |
| author_facet | Tu Yuanyuan Wang Jun |
| author_sort | Tu Yuanyuan |
| collection | DOAJ |
| description | In this article, we are concerned with the existence, non-existence, and blow-up behavior of normalized ground state solutions for the mass critical Hartree-Fock type Schrödinger equation with rotation i∂tu=−Δu+2V(x)u+2ΩLzu−λu−bu∫RN∣u(y)∣2∣x−y∣2dy,(t,x)∈R×RN,u(0,x)=u0(x),\left\{\begin{array}{l}i{\partial }_{t}u=-\Delta u+2V\left(x)u+2\Omega {L}_{z}u-\lambda u-bu\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{{| u(y)| }^{2}}{{| x-y| }^{2}}{\rm{d}}y,\hspace{1em}\left(t,x)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\left(0,x)={u}_{0}\left(x),\hspace{1.0em}\end{array}\right. where N≥3N\ge 3, b>0b\gt 0, V(x)=∣x∣22V\left(x)=\frac{{| x| }^{2}}{2}, Lz{L}_{z} is the angular momentum operator with the critical rotational speed Ω=1\Omega =1, and the constant λ\lambda is the unknown Lagrange multiplier. We prove that the L2{L}^{2}-constraint minimizers exist if and only if the parameter bb satisfies b<b*=‖U‖22b\lt {b}_{* }={\Vert U\Vert }_{2}^{2}, where UU is a positive radially symmetric ground state of −Δu+u−u∫RNu2(y)∣x−y∣2dy=0-\Delta u+u-u{\int }_{{{\mathbb{R}}}^{N}}\frac{{u}^{2}(y)}{{| x-y| }^{2}}{\rm{d}}y=0 in RN{{\mathbb{R}}}^{N}. We also establish the orbital stability result of prescribed mass standing waves for the equation when b<b*b\lt {b}_{* }. When bb approaches b*{b}_{* }, the system collapses to a profile obtained from the optimizer of a Gagliardo-Nirenberg inequality. |
| format | Article |
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| language | English |
| publishDate | 2025-06-01 |
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| series | Advances in Nonlinear Analysis |
| spelling | doaj-art-0f94c4d80ec4416ca764c8ce3dd5f7592025-08-20T02:08:12ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2025-06-011416066607010.1515/anona-2025-0085Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speedTu Yuanyuan0Wang Jun1School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu, 212013, P.R. ChinaSchool of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu, 212013, P.R. ChinaIn this article, we are concerned with the existence, non-existence, and blow-up behavior of normalized ground state solutions for the mass critical Hartree-Fock type Schrödinger equation with rotation i∂tu=−Δu+2V(x)u+2ΩLzu−λu−bu∫RN∣u(y)∣2∣x−y∣2dy,(t,x)∈R×RN,u(0,x)=u0(x),\left\{\begin{array}{l}i{\partial }_{t}u=-\Delta u+2V\left(x)u+2\Omega {L}_{z}u-\lambda u-bu\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{{| u(y)| }^{2}}{{| x-y| }^{2}}{\rm{d}}y,\hspace{1em}\left(t,x)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\left(0,x)={u}_{0}\left(x),\hspace{1.0em}\end{array}\right. where N≥3N\ge 3, b>0b\gt 0, V(x)=∣x∣22V\left(x)=\frac{{| x| }^{2}}{2}, Lz{L}_{z} is the angular momentum operator with the critical rotational speed Ω=1\Omega =1, and the constant λ\lambda is the unknown Lagrange multiplier. We prove that the L2{L}^{2}-constraint minimizers exist if and only if the parameter bb satisfies b<b*=‖U‖22b\lt {b}_{* }={\Vert U\Vert }_{2}^{2}, where UU is a positive radially symmetric ground state of −Δu+u−u∫RNu2(y)∣x−y∣2dy=0-\Delta u+u-u{\int }_{{{\mathbb{R}}}^{N}}\frac{{u}^{2}(y)}{{| x-y| }^{2}}{\rm{d}}y=0 in RN{{\mathbb{R}}}^{N}. We also establish the orbital stability result of prescribed mass standing waves for the equation when b<b*b\lt {b}_{* }. When bb approaches b*{b}_{* }, the system collapses to a profile obtained from the optimizer of a Gagliardo-Nirenberg inequality.https://doi.org/10.1515/anona-2025-0085hartree-fock type schrödinger equationnormalized solutionsrotationblow-up solutions35j6135j2035q5549j40 |
| spellingShingle | Tu Yuanyuan Wang Jun Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed Advances in Nonlinear Analysis hartree-fock type schrödinger equation normalized solutions rotation blow-up solutions 35j61 35j20 35q55 49j40 |
| title | Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed |
| title_full | Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed |
| title_fullStr | Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed |
| title_full_unstemmed | Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed |
| title_short | Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed |
| title_sort | blow up solutions to the hartree fock type schrodinger equation with the critical rotational speed |
| topic | hartree-fock type schrödinger equation normalized solutions rotation blow-up solutions 35j61 35j20 35q55 49j40 |
| url | https://doi.org/10.1515/anona-2025-0085 |
| work_keys_str_mv | AT tuyuanyuan blowupsolutionstothehartreefocktypeschrodingerequationwiththecriticalrotationalspeed AT wangjun blowupsolutionstothehartreefocktypeschrodingerequationwiththecriticalrotationalspeed |