Formation of radial patterns via mixed attractive and repulsive interactions for Schrödinger systems

This paper is concerned with the asymptotic behavior of least energy vector solutions for nonlinear Schrödinger systems with mixed couplings of attractive and repulsive forces. We focus here on the radially symmetric case while the general studies were already conducted in our earlier work [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Math. Pures Appl. (9), 106 (2016), pp. 477–511], [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Fixed Point Theory Appl., 19 (2017), pp. 559–583]. Though there is still the general phenomenon of component-wise pattern formation with co-existence of partial synchronization and segregation for positive least energy vector solutions as in [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Math. Pures Appl. (9), 106 (2016), pp. 477–511], [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Fixed Point Theory Appl., 19 (2017), pp. 559–583], in our case of radially symmetric domains, it turns out that the energy of synchronization part may be concentrated either on the center of the domain or on the boundary of the domain depending on the spatial dimension of the domain. This is a distinct new feature from [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Math. Pures Appl. (9), 106 (2016), pp. 477–511], [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Fixed Point Theory Appl., 19 (2017), pp. 559–583] due to the radially symmetric property. Our approach develops techniques of multiscale asymptotic estimates.