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

Abstract

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 $\varepsilon>0$, 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 $q\in [1,\infty]$.

Theorems & Definitions (23)

• Example 1.1
• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Remark 2.1
• Definition 2.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3