Mass Concentration and Asymptotic Uniqueness of Ground State for 3-Component BEC with External Potential in ℝ2

Abstract

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 βi⁢j=βj⁢i{\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 βi⁢j{\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 βi⁢j↗βi⁢j*:=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 Δ⁢w-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.

Keywords

• ground state
• 3-component bose–einstein condensate
• mass concentration
• nondegeneracy
• uniqueness
• 35j50
• 35q40
• 35q55