On least energy solutions for a nonlinear Schrödinger system with $K$-wise interaction

Abstract

In this paper we establish existence and properties of minimal energy solutions for the weakly coupled system $$\{\begin{array}{ll}-\hat{I}”u_i+\hat{I}{»}_i{u}_i=\hat{I}{¼}_i|u_i{|}^{Kq-2}{u}_i+{\hat{I}}^2|u_i{|}^{q-2}{u}_i{\prod }_{j\ne i}|u_j{|}^q& \text{in }{\mathbb{R}}^d,u_i\in H^1({\mathbb{R}}^d),\end{array}i=1,\dots ,K,$$ characterized by $K$-wise interaction (namely the interaction term involves the product of all the components). We consider both attractive (${\hat{I}}^2>0$) and repulsive cases (${\hat{I}}^2<0$), and we give sufficient conditions on ${\hat{I}}^2$ 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 ${\hat{I}}^2\to -\infty$, showing partial segregation phenomena which differ substantially from those arising in pairwise interaction models.

26 pages, no figures