Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime

Simone Dovetta; Angela Pistoia; Simone Dovetta; Angela Pistoia

doi:10.3934/mine.2022027

Mathematics in Engineering

2022, Volume 4, Issue 4: 1-21. doi: 10.3934/mine.2022027

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Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime

Simone Dovetta ,
Angela Pistoia ^,

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Scarpa 16, 00161 Roma, Italy

Received: 01 May 2021 Accepted: 16 August 2021 Published: 19 August 2021

We study the existence of solutions to the cubic Schrödinger system

$ -\Delta u_i = \sum\limits_{j = 1}^m \beta_{ij} u_j^2u_i + \lambda_i u_i\ \hbox{in}\ \Omega,\ u_i = 0\ \hbox{on}\ \partial\Omega,\ i = 1,\dots,m, $

when $ \Omega $ is a bounded domain in $ \mathbb R^4, $ $ \lambda_i $ are positive small numbers, $ \beta_{ij} $ are real numbers so that $ \beta_{ii} > 0 $ and $ \beta_{ij} = \beta_{ji} $, $ i\neq j $. We assemble the components $ u_i $ in groups so that all the interaction forces $ \beta_{ij} $ among components of the same group are attractive, i.e., $ \beta_{ij} > 0 $, while forces among components of different groups are repulsive or weakly attractive, i.e., $ \beta_{ij} < \overline\beta $ for some $ \overline\beta $ small. We find solutions such that each component within a given group blows-up around the same point and the different groups blow-up around different points, as all the parameters $ \lambda_i $'s approach zero.
- cubic Schrödinger system,
- attractive and repulsive forces,
- blow–up phenomenon,
- Ljapunov–Schmidt reduction
Citation: Simone Dovetta, Angela Pistoia. Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime[J]. Mathematics in Engineering, 2022, 4(4): 1-21. doi: 10.3934/mine.2022027

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Abstract

We study the existence of solutions to the cubic Schrödinger system

$ -\Delta u_i = \sum\limits_{j = 1}^m \beta_{ij} u_j^2u_i + \lambda_i u_i\ \hbox{in}\ \Omega,\ u_i = 0\ \hbox{on}\ \partial\Omega,\ i = 1,\dots,m, $

when $ \Omega $ is a bounded domain in $ \mathbb R^4, $ $ \lambda_i $ are positive small numbers, $ \beta_{ij} $ are real numbers so that $ \beta_{ii} > 0 $ and $ \beta_{ij} = \beta_{ji} $, $ i\neq j $. We assemble the components $ u_i $ in groups so that all the interaction forces $ \beta_{ij} $ among components of the same group are attractive, i.e., $ \beta_{ij} > 0 $, while forces among components of different groups are repulsive or weakly attractive, i.e., $ \beta_{ij} < \overline\beta $ for some $ \overline\beta $ small. We find solutions such that each component within a given group blows-up around the same point and the different groups blow-up around different points, as all the parameters $ \lambda_i $'s approach zero.

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