On least energy solutions for a nonlinear Schrödinger system with K-wise interaction

In this paper we establish existence and properties of minimal energy solutions for the weakly coupled system −Δui+λiui=μi|ui|Kq−2ui+β|ui|q−2ui∏j≠i|uj|qinRdui∈H1(Rd),i=1,…,K,characterized by K-wise interaction (namely the interaction term involves the product of all the components). We consider both attractive (β>0) and repulsive cases (β<0), and we give sufficient conditions on β in order to have least energy fully non-trivial solutions, if necessary under a radial constraint. We also study the asymptotic behaviour of least energy fully non-trivial radial solutions in the limit of strong competition β→−∞, showing partial segregation phenomena which differ substantially from those arising in pairwise interaction models.