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

Sabrina Caputo

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.

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

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.

Paper Structure

This paper contains 7 sections, 9 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. The finite dimensional reduction
  3. Setting of the problem
  4. The invertibility of linear operator $\mathcal{L}$
  5. The size of the error term
  6. The reduced problem
  7. Useful estimates

Key Result

Proposition 1.1

Let $t, r >1$ and $0<\mu<N$ with $\frac{1}{t}+\frac{\mu}{N}+\frac{1}{r}=2.$ There exists a sharp constant $C=C(t,N,\mu,r)$ such that for any $f\in L^t({\mathbb{R}^N})$ and $h\in L^r({\mathbb{R}^N})$ $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (15)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Proposition 2.6
  • ...and 5 more