We are concerned with the following system of two coupled time-independent Gross-Pitaevskii equations cases -Δu+λ₁ u=μ₁|u|^p-2u+να|u|^α-2|v|^βu ~in~ N, \\ -Δv+λ₂ v=μ₂|v|^q-2v+νβ|u|^α|v|^β-2v ~in~ N, cases which arises in two-components Bose-Einstein condensates and involve attractive Sobolev subcritical or critical interactions, i. e. , ν>0 and α+β 2^*. This system is employed by seeking critical points of the associated variational functional with the constrained mass below ₑ₍|u|² dx=a, ₑ₍|v|² dx=b. In the mass mixed case, i. e. , 2<p<2+4N<q<2^*, for some suitable a, b, ν and β, the system above admits two positive solutions. In particular, in the case α+β<2^*, using variational methods on the L²-ball, two positive solutions are obtained, one of which is a local minimizer and the second one is a mountain pass solution.
Zhang et al. (Thu,) studied this question.