Fully sign-changing Nehari constraint vs sign-changing solutions of a competitive Schrödinger system

We study a competitive nonlinear Schrödinger system in ${\mathbb{R}}^N$ whose nonlinear potential is localized in small regions that shrink to isolated points. Within a variational framework based on a fully sign-changing Nehari constraint and Krasnosel’skii genus, we construct, for all $\epsilon >0$, a sequence of sign-changing solutions with increasing and unbounded energies, and after suitable translations they converge to a sequence of sign-changing solutions of the associated limiting system as $\epsilon \to 0$ in $H^1$-norm. Moreover, these sign-changing solutions concentrate around the prescribed attraction points both in $H^1$-norm and $L^q$-norm for $q\in [1,\mathrm{\infty }]$.