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

Abstract

Abstract We consider the following elliptic system $$\begin{aligned} -\Delta u_i=(|x|^{-\mu }*|u_i|^{4-\frac{\mu }{2}})|u_i|^{2-\frac{\mu }{2}}u_i+\sum _{j=1\, j\not =i}^2\beta u_iu_j^2\quad \hbox {in} \, \, {\mathbb {R}^4},\quad i=1,2. \end{aligned}$$ - Δ u i = ( | x | - μ ∗ | u i | 4 - μ 2 ) | u i | 2 - μ 2 u i + ∑ j = 1 j ≠ i 2 β u i u j 2 in R 4 , i = 1 , 2 . If $$\beta <0, |\beta |$$ β < 0 , | β | is small enough we build solutions such that the first component looks like the radial positive solution of the single equation, while the second one blows-up at the k vertices of a regular polygon.