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 $$\{\begin{array}{ll}-\Delta u_i+{\lambda }_i{u}_i={\mu }_i|u_i{|}^{Kq-2}{u}_i+\beta |u_i{|}^{q-2}{u}_i{\prod }_{j\ne i}|u_j{|}^q& \text{in }{\mathbb{R}}^d,u_i\in H^1({\mathbb{R}}^d),\end{array}i=1,\dots ,K,$$ characterized by $K$-wise interaction (namely the interaction term involves the product of all the components). We consider both attractive ($\beta >0$) 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.

arXiv comment26 pages, no figures

References in corpus (4)

• Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case
• Multipulse phases in k-mixtures of Bose-Einstein condensates
• Three-Body Interacting Bosons in Free Space
• On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition