SYMMETRY AND ASYMPTOTIC BEHAVIOR OF GROUND STATE SOLUTIONS FOR SCHRODINGER SYSTEMS WITH LINEAR INTERACTION

We study symmetry and asymptotic behavior of ground state solutions for the doubly coupled nonlinear Schrodinger elliptic system { -Delta u + lambda(1)u+ kappa v = mu(1)u(3) + beta uv(2) in Omega, -Delta v + lambda(2)v + kappa u = mu(2)v(3) + beta u(2)v in Omega, u = v = 0 on partial derivative Omega (or u, v is an element of H-1(R-N) as Omega = R-N,R- where N <= 3,Omega subset of R-N is a smooth domain. First we establish the symmetry of ground state solutions, that is, when Omega is radially symmetric, we show that ground state solution is foliated Schwarz symmetric with respect to the same point. Moreover, ground state solutions must be radially symmetric under the condition that Omega is a ball or the whole space R-N. Next we investigate the asymptotic behavior of positive ground state solution as k -> 0(-) , which shows that the limiting profile is exactly a minimizer for c(0) (the minimized energy constrained on Nehari manifold corresponds to the limit systems). Finally for a system with three equations, we prove the existence of ground state solutions whose all components are nonzero.