Multiple normalized solutions for two coupled Gross-Pitaevskii equations with attractive interactions and mass constriants

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 ∫RN|u|² dx=a, ∫RN|v|² dx=b. In the mass mixed case, i. e., 2<p<2+4/N<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.