In this paper we establish existence and properties of minimal energy solutions for the weakly coupled system $$ \begin{ cases} -\Delta u_i + \lambda_i u_i = \mu_i|u_i|^{ Kq-2}u_i + \beta|u_i|^{ q-2}u_i\prod_{ j\neq i}|u_j|^q & \text{ in }\mathbb{ R}^d, \qquad u_i \in H^1(\mathbb{ R}^d), \end{ cases}\qquad i=1,\dots, K, $$ 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 $\beta \to -\infty$, showing partial segregation phenomena which differ substantially from those arising in pairwise interaction models.
Categories: math.AP