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 $$\{\begin{array}{ll}-\mathrm{\Delta }{u}_i+{\lambda }_i{u}_i={\mu }_i\mathrm{\mid }{u}_i{\mathrm{\mid }}^{Kq-2}{u}_i+\beta \mathrm{\mid }{u}_i{\mathrm{\mid }}^{q-2}{u}_i{\prod }_{j\mathrm{\ne }i}\mathrm{\mid }{u}_j{\mathrm{\mid }}^q& \text{in }{\mathbb{R}}^d,u_i\in H^1({\mathbb{R}}^d),\end{array}i=1,\dots ,K,$$ characterized by $K$-wise interaction (namely the interaction term involves the product of all the components). We consider both attractive ($\beta >0$) and repulsive cases ($\beta <0$), and we give sufficient conditions on $\beta$ in order to have least energy fully non-trivial solutions, if necessary under a radial constraint. We also study the asymptotic behavior of least energy fully non-trivial radial solutions in the limit of strong competition $\beta \to -\mathrm{\infty }$, showing partial segregation phenomena which differ substantially from those arising in pairwise interaction models.