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arXiv · 2507.03480

On least energy solutions for a nonlinear Schr\"odinger system with $K$-wise interaction

Lorenzo Giaretto·Nicola Soave
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Abstract

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

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Lorenzo Giaretto, Nicola Soave. 2025-07-04. On least energy solutions for a nonlinear Schr\"odinger system with $K$-wise interaction. https://doi.org/10.1016/j.na.2025.113938

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