preprint

Inserted: 23 jul 2025

Year: 2025

Abstract:

In this paper we establish existence and properties of minimal energy solutions for the weakly coupled system $$ \begin{cases} -\Delta u_i + \lambda_i u_i = \mu_i
ui
^{Kq-2}u_i + \beta
u_i
^{q-2}u_i\prod{j\neq i}
u_j
^q & \text{in }\mathbb{R}^d, \qquad u_i \in H¹(\mathbb{R}^d), \end{cases}\qquad i=1,\dots, K, $$ 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.