2020 Mathematics Subject Classification. Primary: 35B44, 35J47; Secondary: 35B33. Key words and phrases. Gross-Pitaevskii systems, segregated solutions, blowing-up solutions, critical exponent. H. Chen is partially supported by [the NSFC grants (No.12071169)] and [the China Scholarship Council (No. 202006770017)]. The second and the third authors are partially supported by [INDAM-GNAMPA funds]. A. Pistoia is also partially supported by [Fondi di Ateneo “Sapienza” Università di Roma (Italy)]. G. Vaira is also partially supported by [PRIN 2017JPCAPN003 “Qualitative and quantitative aspects of nonlinear PDEs”]. ∗Corresponding author: Angela Pistoia.