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 $\left(\begin{array}{c}-\Delta u_i+{\lambda }_i{u}_i={\mu }_i|u_i{|}^{Kq-2}{u}_i+\beta |u_i{|}^{q-2}{u}_i\prod _{j\ne i}|u_j{|}^q\text{in}{R}^d\\ u_i\in H^1\left(R^d\right),\end{array}\right)i=1,\dots ,K,$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 behaviour of least energy fully non-trivial radial solutions in the limit of strong competition $\beta \to -\infty$, showing partial segregation phenomena which differ substantially from those arising in pairwise interaction models.