Fully sign-changing Nehari constraint vs sign-changing solutions of a competitive Schr\"{o}dinger system

TL;DR

This paper constructs sign-changing solutions for a competitive Schrödinger system with localized nonlinear potentials, showing their concentration behavior and convergence to solutions of a limiting system as the localization shrinks.

Contribution

It introduces a variational approach using a fully sign-changing Nehari constraint to find multiple sign-changing solutions with unbounded energies.

Findings

Solutions concentrate around attraction points in H^1-norm.

Solutions also concentrate in L^q-norm for q in [1,∞].

Constructs solutions for all ε>0 with increasing energies.

Abstract

We study a competitive nonlinear Schr\"odinger 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 $\varepsilon>0$, 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 $\varepsilon\to 0$ 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 $q\in [1,\infty]$.