Asymptotic behaviors of least energy solutions for weakly coupled nonlinear Schrödinger systems

Abstract We study the asymptotic behavior of positive least energy solutions to the weakly coupled nonlinear Schrödinger systems with nearly critical exponents. There are at most three kinds of asymptotic behaviors for positive solutions to two-coupled systems: (i) both two components converge; (ii) one component blows up while the other one converges; (iii) both two components blow up. Here we prove that any type of these three asymptotic behaviors will happen under different circumstances.