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

Abstract

Abstract We investigate the ground states of 3-component Bose–Einstein condensates with harmonic-like trapping potentials in ℝ 2 {\mathbb{R}^{2}} , where the intra-component interactions μ i {\mu_{i}} and the inter-component interactions β i ⁢ j = β j ⁢ 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 μ i {\mu_{i}} and β i ⁢ 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 β i ⁢ j ↗ β i ⁢ j * := a ∗ + 1 2 ⁢ ( a ∗ - μ i ) ⁢ ( a ∗ - μ j ) {\beta_{ij}\nearrow\beta_{ij}^{*}:=a^{\ast}+\frac{1}{2}\sqrt{{(a^{\ast}-\mu_{i% })(a^{\ast}-\mu_{j})}}} , where 0 < μ i < a ∗ := ∥ w ∥ 2 2 {0<\mu_{i}<a^{\ast}:=\|w\|_{2}^{2}} are fixed and w is the unique positive solution of Δ ⁢ w - w + w 3 = 0 {\Delta w-w+w^{3}=0} in H 1 ⁢ ( ℝ 2 ) {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.