Mass Concentration and Asymptotic Uniqueness of Ground State for 3-Component BEC with External Potential in ℝ2
We investigate the ground states of 3-component Bose–Einstein condensates with harmonic-like trapping potentials in ℝ2{\mathbb{R}^{2}}, where the intra-component interactions μi{\mu_{i}} and the inter-component interactions βij=βji{\beta_{ij}=\beta_{ji}} (i,j=1,2,3{i,j=1,2,3}, i≠j{i\neq j}) are al...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2021-08-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2021-2131 |
| Summary: | We investigate the ground states of 3-component Bose–Einstein condensates with harmonic-like trapping potentials in ℝ2{\mathbb{R}^{2}}, where the intra-component interactions μi{\mu_{i}} and the inter-component interactions βij=βji{\beta_{ij}=\beta_{ji}} (i,j=1,2,3{i,j=1,2,3}, i≠j{i\neq j}) are all attractive. We display the regions of μi{\mu_{i}} and βij{\beta_{ij}} for the existence and nonexistence of the ground states, and give an elaborate analysis for the asymptotic behavior of the ground states as βij↗βij*:=a∗+12(a∗-μi)(a∗-μj){\beta_{ij}\nearrow\beta_{ij}^{*}:=a^{\ast}+\frac{1}{2}\sqrt{{(a^{\ast}-\mu_{i% })(a^{\ast}-\mu_{j})}}}, where 0<μi<a∗:=∥w∥22{0<\mu_{i}<a^{\ast}:=\|w\|_{2}^{2}} are fixed and w is the unique positive solution of Δw-w+w3=0{\Delta w-w+w^{3}=0} in H1(ℝ2){H^{1}(\mathbb{R}^{2})}. The energy estimation as well as the mass concentration phenomena are studied, and when two of the intra-component interactions are equal, the nondegeneracy and the uniqueness of the ground states are proved. |
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| ISSN: | 1536-1365 2169-0375 |