@article{Dovetta2022, 
author = {Simone Dovetta and Angela Pistoia},
title = {Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime},
year = {2022},
journal = {Mathematics in Engineering},
volume = {4},
number = {4},
pages = {1-21},
keywords = {cubic Schrödinger system, attractive and repulsive forces, blow–up phenomenon, Ljapunov–Schmidt reduction},
url = {https://www.sciopen.com/article/10.3934/mine.2022027},
doi = {10.3934/mine.2022027},
abstract = {We study the existence of solutions to the cubic Schrödinger system     −  Δ      u    i    =      ∑          j      =      1        m        β          i      j            u    j    2        u    i    +      λ    i        u    i          in      Ω  ,        u    i    =  0        on      ∂  Ω  ,    i  =  1  ,  …  ,  m  ,when    Ω is a bounded domain in              R        4    ,        λ    i   are positive small numbers,        β          i      j       are real numbers so that        β          i      i        >  0 and        β          i      j        =      β          j      i      ,    i  ≠  j. We assemble the components        u    i   in groups so that all the interaction forces        β          i      j       among components of the same group are attractive, i.e.,        β          i      j        >  0, while forces among components of different groups are repulsive or weakly attractive, i.e.,        β          i      j        <      β    ¯   for some        β    ¯   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        λ    i  's approach zero.}
}