Segregated solutions for a critical Choquard system with a small interspecies repulsive force

TL;DR

This paper constructs multi-bubble segregated solutions for a critical nonlocal Choquard system in four dimensions, revealing how nonlocal interactions influence segregation patterns similar to local systems.

Contribution

It introduces a novel finite-dimensional reduction method to establish the existence of segregated solutions with specific concentration behaviors in a critical nonlocal system.

Findings

Existence of multi-bubble segregated solutions with specific concentration patterns.

Nonlocal terms preserve qualitative segregation patterns observed in local systems.

Solutions include a ground state and blow-up behavior at polygon vertices.

Abstract

In this work, I focus on a coupled system of nonlinear Choquard equations in dimension 4, characterized by critical nonlocal nonlinearities and a small repulsive interspecies interaction. I prove the existence of a new class of multi-bubble segregated solutions. Specifically, I construct solutions where the first component concentrates as a radial positive ground state, while the second component exhibits a blow-up behaviour, concentrating at k points arranged as the vertices of a regular polygon. The proof relies on a sophisticated finite-dimensional reduction method, bridging the gap between the theory of competitive systems and critical nonlocal equations. My results show that the presence of nonlocal terms preserves the qualitative segregation patterns typically observed in local Schrodinger systems.