Symmetric results of a Henon-type elliptic system with coupled linear part

In this article, we study the elliptic system: ( delta u + mu(1)u = | x|(alpha)u(3) + lambda v, x is an element of omega - delta v+mu(2)v = l x l (alpha)v(3 )+ lambda u , x is an element of omega u,v > 0, x epsilon omega. u = v = 0, x is an element of phi omega, where omega subset of R-3 is the unit ball. By the variational method, we prove that if alpha is sufficiently small, the ground state solutions of the system are radial symmetric, and if alpha > 0 is sufficiently large, the ground state solutions are nonradial; however, the solutions are Schwarz symmetry.